References Software see at the beginning of Chaps. 19 and 24. General References [GR1] Abramowitz,M. andl. A. Stegun (eds.), Handbook of Mathematical Functions. 1Oth printing, with corrections. Washington, DC: National Bureau of Standards. 19]2 (also New York: Dover, 1965). tGR2] Cajori, F., History of Mathematics. 5th ed. Reprinted. Providence, RI: American Mathematical Society, 2002. tGR3] Courant, R. and D. Hilbert, Methods of Mathematical Physics. 2 vols. Hoboken, NJ: Wiley, 2003. IGR4] Courant, R., Dffirential and Integral Calculus. 2 vols. Hoboken, NJ: Wiley, 2003. tGR5] Graham, R. L. et al,, Concrete Mathematics.2nd ed. Reading, MA: Addison-Wesley, 1994. [GR6] Ito, K. (ed.), Encyclopedic Dictionary of Mathematics. 4 vois. 2nd ed. Cambridge, MA: MIT Press, 1993. tGR7] Kreyszig, E., Introductory Functional Analysis with Applications, New York: Wiley, 1989. tGR8] Kreyszig, E., Dffirential Geomerry. Mineola, NY: Dover, 1991. tGR9] Kreyszig, E, Introduction to Dffirential Geometry and Riemannian Geometry. Toronto: University of Toronto Press, 1975. IGR10] Szegó, G., Orthogonal Polynomials. 4th ed. Reprinted. New York: American Mathematical Society, 2003. [GRl1] Thomas, G. et al., Thomas' Calculus, Early Transcendental,s (Ipdate, 1Oth ed. Reading, MA: Addison-Wesley, 2003. Part A. Ordinary Differential Equations (ODEs) (Chaps. 1-6) See also Part E: Numeric Analysis [A1] Arnold, V. L, Ordinaryl Dffirential Equations.3rd, ed. New York: Springer, 1997, [A2] Bhatia, N. P. and G. P. Szego, Stability Theory of Dynamical Systems. New York: Springer, 2002, [A3] Birkhofl G. and G.-C. Rota, Ordinary Dffirential Equations,4th ed. New York: Wiley, 1989. [A4] Brauer, F. and J. A. Nohel, Qualitative Theoryl of Ordinary Differential Equations. Mineola, NY: Dover, 1994. [A5] Churchill, R. Y., Operational Mathematics.3rd ed. New York: McGraw-Hlll, 1972. ['{6] Coddington, E. A. and R. Carlson, Linear Ordinary Dffirential Equations. Philadelphia: SIAM, I9g7. [A7] Coddington, E. A. and N. Levinson, Theory of Ordinaryl Dffirential Equations. Malabar, FL: Krieger, 1984. [A8] Dong, T.-R. etal., Qualitative Theory of Dffirential Equations. Providence, RI: American Mathematical Society, 1992. [A9] Erdélyi, A. et a1., Tables of Integral Transforms. 2 vols. New York: McGraw-Hill, 1954. tA10] Hartman, P., Ordinary Differential Equations.2nd ed. Philadelphia: SIAM, 2002. [A11] Ince, E. L,, Ordinary Dffirential Equations. New York: Dover, 1956. LAIZ) Schiff, L L., The Laplace Transform: Theory and Applications. New York: Springer, 1999. tAi3] Watson, G. N., A Treatise on the Theory of Bessel Functions. 2nd ed. Reprinted. New York: Cambridge University Press, 1995. tA14] Widder, D. V., The Laplace Transform. Princeton, NJ: Princeton University Press, 1941. [A15] Zwillinger,D., Handbook of Diftbrential Equations. 3rd ed. New York: Academic Press, 1997. Part B. Linear Algebra, Vector Calculus (Chaps. 7-10) For books on numeric linear algebra, see also Part E: Numeric Analysis. [B1] Bellman, R., Introduction to Matrix Analysis. 2nd, ed. Philadelphia: SIAM, 1997. [B2] Chatelin, F., Eigenvalues of Matrices. New York: Wiley-Interscience, 1993. [B3] Gantmacher, F. R., The Theory of Matrices. 2 vols, Providence, RI: American Mathematical Society, 2000. [B4] Gohberg,I. P. et al., Invariant Subspaces of Matrices with Applications. New York: Wiley, 1986. [B5] Greub, W. H., Linear Algebra.4th ed. New York: Springer, 1996. [86] Herstein, I. N., Abstract Algebra.3rd ed. New York: Wiley, 1996. [B7] John, A. W., Maírices and Tensors in Physics. 3rd, ed. New York: Wiley, 1995. At APPENDlX l References [B8] Lang, S., Linear Algebra, 3rd ed. New York: Springer, 1996. [B9] Nef, -V,I., Linear Algebra.2nd ed, New York: Dover, 1988. tB10] Parlett, B., The Symmetric Eigenvalue Problem. Philadelphia: SIAM, 1997. Part C. Fourier Analysis and PDEs (Chaps. 10-1l) For books on numerics for pDEs see also part E: Numeric Analysis. [C1] Antimirov, M. Ya., Applied Inte7ral Transforms, Providence, RI: American Mathematical Society, 1993. [C2] Bracewell, R., The Fowrier Transform and lts Applications, 3rd ed. New York: McGraw-Hill, 2000. [C3] Carslaw, H. S. and J, C. Jaeger, Conduction of Heat in Solids, 2nd ed. Reprinted. Oxford: Clarendon, 1986. [C4] Churchill, R. V. and J. W. Brown, Fourier Series and Boundary Value Problems. 6th ed. New York: McGraw-Hill, 2000. [C5] DuChateau, P. and D. Zachmann, Applied Particll Dffirential Eqwations. Mineola, NY: Dover, Z00I. [C6] Hanna, J. R. and J. H. Rowland, Fourier Series, Transforms, and Boundary Valwe Problems. 2nd ed. New York: Wiley, 1990. [C7] Jerri, A. J., The Gibbs Phenomenon in Fourier Analy s i s, Sp l ine s, and W av e le t Ap p r o ximrlt io ns. B oston : Kluwer, 1998. [C8] John, F., Partial Dffirential Equations. New York: Springer, 1995. [C9] Tolstov, G. P., Fourier Series. New York: Dover, I916. tC10] Widder, D, V., The Heat Equation. New York: Academic Press, 1975. [C11] Zauderer, E., Partial Dffirential Equations oJ Applied Mathematics.2nď ed. New York: Wiley, 1989. |CI2] Zygmund, A. and R. Fefferman, Trigonometric Series.3rd ed. New York: Cambridge University Press, 2003. Part D. Complex Analysis (Chaps. 13-18) [D1] Ahlfors, L. Y., Complex Analysis. 3rd ed. New York: McGraw-H1II, 19] 9. [D2] Bieberbach, L., Conformal Mapping. Providence, RI: American Mathematical Society, 2000. [D3] Henrici, P., Applied and Computational Complex Analysis,3 vols. New York: Wiley, 1993. [D4] Hille, E., Analytic Function Theory. 2 vols. 2nd ed. Providence, RI: American Mathematical Society, 1997. [D5] Knopp, K., Elements of the Theory of Functions. New York: Dover, 1952. [D6] Knopp, K., Theoryl of Functions. 2 parts. New York: Dover, 1945, 1941. [D7] Krantz, S. G., Complex Analysis: The Geometric Viewpoint. Washington, DC: The Mathematical Association of America, l990. [D8] Lang, S., Complex Analysis. 3rd ed. New York: Springer, l993. [D9] Narasimhan, R., Compact Riemann,Sufaces. New York: Springer, 1996. tD10] Nehari, Z., Conformal Mapping, Mineola, NY: Dover. 1975. [Dll] Springer, G., Introduction to Riemann Surfaces, Providence, RI: American Mathematical Society, 2002. Part E. Numeric Analysis (Chaps. 19-21) [E1] Ames, W. F., Numerical Methods for Partial Dffirentictl Equations, 3rd ed. New York: Academic Press. 1992, [E2] Anderson, E., et al., LAPACK User's Guide.3rd ed. Philadelphia: SIAM, 1999. [E3] Bank, R. E., PLTMG, A Software Package for Solving Elliptic Partial DffirentiaL Equations: Users' Guide 7.0. Philadelphia: SIAM, 1994. [E4] Constanda, C., Solution Techniques .for ElementcLty Partial Differential Equations. Boca Raton, FL: CRC Press, 2002. [E5] Dahlquist, G. and A. Bjórck, Numerical Methocls. Mineola, NY: Dover, 2003. [E6] DeBoor, C., A Practical Guide to Splines. Reprinted. New York: Springer, l991, [E7] Dongarra, J. J. et al., LINPACK Users Guide. Philadelphia: SIAM, 1978. (See also at the beginning of Chap. 19.) [E8] Garbow, B. S. et al., Matrix Eigensystem Routines: EISPACK Guide Extension. Reprinted. New York: Springer, l990. [E9] Golub, G, H, and C. F. Van Loan, Matrix Computatiolrs. 3rd ed. Baltimore, MD: Johns Hopkins University Press, 1996. [E10] Higham, N. J., Accuracy and Stability of Numerical Algorithms. 2nd ed. Philadelphia: SIAM, 2002. [El1] IMSL (International Mathematical and Statistical Libraries), FORTRAN Numerical Library. Houston, TX: Visual Numerics, 2002. (See also at the beginning of Chap. l9.) [El2] IMSL, IMSL for Java. Houston, TX: Visual Numerics, 2002. [E13] IMSL, C Library. Houston, TX: Visual Numerics, 2002. [E14] Kelley, C. T,, Iterative Methods .for Linear cLnd Nonlinear Equations. Phitadelphia: SIAM, 1995. [E15] Knabner, P. and L. Angerman, Numerical Methods for Partial Dffirential Equations. New York: Springer, 2oo3. [El6] Knuth, D. E., The Art of Computer Programming. 3 vols. 3rd ed. Reading, MA: Addison-Wesley, 2005. --4, App.l [E17] Kreyszig, E., Introductoty Functional Analysis with Applications. New York: Wiley, 1989. [E18] IGeyszig, E., On methods of Fourier analysis in multigrid theory. Lecture Notes in Pure and Applied Mathematics 1 57. New York: Dekker, I994, pp. 225-242. [E19] Kreyszig, E., Basic ideas in modern numerical analysis and their origins. Proceedings of the Annuctl Conference of tlle Canadian Society for the History and Philosophy of Mathematics. 199], pp. 34*45, [E20] Ifueyszig, E., and J. Todd, QR íntwo dimensions. Elemente der Mathematik 3l (1976), pp, 109-114. [E21] Mortensen, M. E., Geometric Mocleling. 2nd ed, New York: Wiley, 1997. |E22] Morton, K. W., and D. F. Mayers, Numerical Solution of Partial Dffirential Equctions; An Introduction. New York: Cambridge University Press, 1994. [E23] Ortega, J. M., Introduction to Parallel ancl Vector Solution of Linear Systems. New York: Plenum Press, 1988. [E24l Overton, M. L., Numerical Computing witlt IEEE Floating Point Arithmetic. Philadelphia: SIAM, 20OI. [E25] Press, W. H. et al., Nunterical Recipes in C: The Art of Scientific Computittg,2nd, ed. New York: Cambridge University Press, l992. [E26] Shampine, L. F., Numerical Solutions of Orclinary Differential Equations. New York: Chapman and Hall, 1994. [E27l Yarga, R. S., Matrix lterative Analysis.2nd ed. New York: Springer, 2000. tE28] Varga, R. S., Geršgorin and His Circles. New York: Springer, 2004. |E29l Wilkinson, J. H., The Algebraic Eigenvalue Problem, Oxford: Oxford University Press, i988, Part F. Optimization, Graphs (Chaps. 22-23| [F1] Bondy, J. A., Graph Theory with Apptications. Hoboken, NJ: Wiley-Interscienc e, 2a03, [F2] Cook, W. J. et al., Combinatorial Optimizatiolr. New York: Wiley, 1993. [F3] Diestel, R., Graph Theory. 2nd ed. New York: Springer, 2000. [F4] Diwekar, IJ. M., Introduction to Applied Optilnization. Boston: Kluwer, 2003. [F5] Gass, S. L., Linear Programming. Method and Applications. 3rd ed. New York: McGraw-Hill. 1969. [F6] Gross, J. T., Handbook of Graph Theoryl ancl Appliccttions. Boca Raton, FL: CRC Press, i999. [F7] Goodrich, M. T., and R. Tamassia, Algorithtn Design: Foundations, AncLlysis, and Internet Examples. Hoboken, NJ: Wiley, 2002. [F8] Harary,F., Graph Theory. Reprinted. Reading, MA: Addison-Wesley, 2000. [F9] Merris, R., Graph Theoryl. Hoboken, NJ: WileyInterscience. 2000. [F10] Ralston, A,, and P. Rabinowitz, A First Course in Numerical Analysis.2nd ed. Mineola, NY: Dover ,2001, [F11] Thulasiraman, K., and M. N. S. Swamy, Graph Theory and Algorithms. New York: Wiley-Interscience, 1992. [F12] Tucker, A., Applied Combinatorics. 4th ed. Hoboken, NJ: Wiley, 200L Part G. Probability and Statistics (Chaps. 24-25I [G1] American Society for Testing Materials, Mantral on Presentation of Data and Control Chart Analysis. 7th ed. Philadelphia: ASTM, 2002. [G2] Anderson, T. W., An Introduction to Multivariate Statistical Analysis,3rd ed. Hoboken, NJ: Wiley, 2003. [G3] Cramér, H., Mathematical Methods of Statistics. Reprinted. Princeton, NJ: Princeton University Press, 1999. [G4] Dodge, Y., The Oxford Dictionary of Stcttistical Terms.6th ed. Oxford: Oxford University Press, 2003. [G5] Gibbons, J. D., Nonparametric Statisticctl Inference. 4th ed. New York: Dekker, 2003. [G6] Grant, E. L. and R. S. Leavenw ofih, Statistical Quality Control. 7th ed. New York: McGraw-Hill, 1996, [G7] IMSL, Fortran Numerical Library, Houston, TX: Visual Numerics, 2002. [G8] Kreyszig, E., Introcluctotry MathetnaticcLl Statistics. Principles and Methorls. New York: Wiley, 1970. [G9] O'Hagan, T. et al., Kendctll's Ad.vanced Theory of Statistics 3-Volume , e/. Kent, U.K.: Hodder Arnold, 2004. tG10] Rohatgi, V. K. and A. K. MD. E. Saleh, Án Introduction to Probability and Statistics. 2nd, ed. Hoboken, NJ: Wiley-Interscience, 2001. APPE, NDlx2 Answers to Odd-Numbered Problems Problem Set 1.1, page 8 1. (cos rrx)lrr -l c 3. e*'lz + c 5. First order 7. Second order 9. Third order Lt.y : ltan (2x + nrr), n : 0, tl, *2,,,, 13. y : e-*' 15. (A) No. (B) No. Only y : 0. 17.y" :8,!' :1t,y: g 12 19.y" : k,!' : kt + 6,y:!t< + ,y(60): 1800k + 360:3000, k: L4J, y'(6o) : I.47, 60 + 6 : 94 [m/sec] : 2I0 [mph] 21,. ekH : ž,H:(ln ž)tt : (10" In2)lI.4 : I5]0 [years] Problem Set 1.2, page 11 11. y : -(2ln) costrrx -l c 15.y:x(I_lnx)*c 17. Verify the general solution y2 + t2 : c. Circle of radius 3\/' 19.ntu' - *g - b,J',u' :9.8 - u2, u(0) : l0. u' :0 gives the limit \, Á: 3.1 [meter/sec]. Problem Set 1.3, page 18 3. cos 2y dy:2dx,y:+arcsin (4x + c) 5.y'+ 36x2: c, ellipses 7. dyly: cot Trx dx,y : c(sínrrx)ll- 9.y : tan (c - e-n*lr) ll.r:ťo-t' 13. I:Ioe-Rttt 15.y:n"lluffia5 17.y:4lnx 19.y:@ 21.y' : (} - b)l(x - a),y - b: c(x - a) 23. ygek : 2!o, ek : 2 (1 week), e2k : 22 (2 weeks), e4k : 24 25.y: lo kt : yo -0,0OOl273t : lo -0,0OO1213'40OO :0.62yg;62%o; cf. Example 2. 27.y' -- -ldy,! : !oe-kt, e-sk: 0.5, k: -(ln 0.5)/5 : 0.139, t - -(ln 0.05)/0.139 : 22 [min] 29.rQ): 10, T:23 - l3ekt,T(2):23 - I3e2k: 18, k -- -0.478,T:22.8 gives , : [ln (-0.2l- 13)]/(-0.478) : 8.73 [min]. 31. h: 8 /2, t : x/Íhtg,u : gt : gÝZt lg : \/kh 33.y' - 0 - (2/800)y, y:20Oe-o,oo25t, t:300 [min],y(300):94.5 [1b] 35. (A) is related to the error function and (C) concerns the Fresnel integral C(x); see App. 3.1. (D) y' : 2xy * 1, y(0) : 0 A4 App. 2 Answers to Odd-Numbered Problems Problem Set 1.4, page 25 1. Exact. xa + y4 : c 3.Exact, u: cos z,xsinh y + k(y),tta: cos zrxcosh y + k',k' :0. Áns. cos zx sinh y : c 5. Exact. 9x2 + 4y' : , 7.Exact,Me: Nr: -2n-'u,Ll: re-2'+ k(OD,uu: -2re-20 + k',k' : g. Ans.re-z'-r,r:Ce2' 9.Exact. u: ylx * sin 2x -| k(y),uu: Ilx l k' : Ilx - 2sin2y, Ans. ylx * sin 2x -| cos 2y : c 11. Not exact. F : llx2 by Theoreml. -y/x'dx + llx dy : d(.ylx): O. Ans. y : cx 13. -3y2lxn dx + 2ylx3 dy : d(y2lx31 : 0. y - cx3l2 (semicubica1 parabolas) 15. Exact, L,l : e2'cos y + k(y), uu : -e2" siny + k', k' : 0. Ans. ez*cos y : c, c:I 17. Not exact. Try R. F : e-*, e-'(cos alx * ro sin rrx) dx + dy: O, u: y -l l(x), ur: l': e-"(cos ax* rr;sin ox),w:y + l:y - e-r cos oJ: C,C:O I9.u: e* + k(y),uu: k' : -1 + ea,k: -y * ea,Ans. e* - y l ea : c 2I. B : C,*Ax2 + Cxy + !Dy2 :, A5 Problem Set 1.5, pate 32 3.y: rr-3,5r +0.8 7.y : x l c (|f k :0).y : ce-k* + 9. Separate. y - 2.5 : c cosha 1.5x 13. y : sin 2x -| clsinz 2x, c : I 5.y: 2.6r-7,z51 14 ezk"l3kifk+0 11.y - 2*ncos2r 15. y - n1l*7x2 * c), c : 4.I : 5 sinh 10x.I7. y: (c -| } cosh 10r)/.r3. Note (x3y)' 19. 1, : llu. u - r,r-5,7,t - -5.1 2l. u : y-2 : e*'(I + ce2'), c : 3, u(0) : 4 23. Separate variables. y2 - 1 - ce"o" *, c : -Ile 25.y' : Ry l k,y: ceRt - klR,c: lo+ klR.yo: 1000,R:0.06, t:65 - 25:40, k: 1000,!: $178076.12. Start at45 gives yo[(l + 1/0.06)eo,o6,20 - 1/0.06] : 4I.988732yo: I78016.12,yo: k: §4240.05. 27. y' : 175(0.0001 - y/450),y(0) : 450.0.0004 : 0.18, y : 0.135r-o,saa9t + 0.045 : 0.78/2, e-osaa%: (0.09 - 0.045)/0.135 : ll3, 1 : (ln 3)/0.3889 : 2.82. Ans. About 3 years 29.y' - A - lq,y(0):0,y : A(I - ,-t"t;/k 31.y' : Byz - Ay: By(y - AlB),A>O,B > 0. Constantsolutionsy:0, y: AlB. y'> 0if y > A/B (unlimitedgrowth), y' <0if 0 (y< AlB(extinction). y : Al(ceo' + B),y(O) > AlB f c ( 0, y(0) < AlB if c ) 0. 33.y' - y - y2 - 0.2y,y: Il(I.25 - g.75r-o,st), tmitO.B, limit 1 35. y' - y - 0.25y2 - 0.1y : 0.25y(3.6 - y). Equilibrium harvest 3.6, y:18/(5+ce-o,st) 37.(yl,+ y)' + p(yr+ y): (y1,' + py) + (yz' + py):0 + 0:0 39. (yr 1- yr)' + pjt + yr): (yr' + pyr) + (y2' + py) : r l 0 : r 41. Solution of cyt' l pclt: c(yr' + py) : cr A6 App. 2 Answers to Odd-Numbered Problems 43. CAS Experiment (a) y: x sin (1/x) l cx. c : 0 if y(2lrr):2lrr. y is undefined at í : 0, the point at which the "waves" of sin (1/x) accumulate; the factor x makes them smaller and smaller. Experiment with various x-intervals. (b)y: x'[sin (1/x) + c].y(2lrr): (2lrr)n.nneed,notbe aninteger.Try n:t. Try n : - 1 and see how the "waves" near 0 become larger and larger. 45.y: u!*,y' * py - u'y* * uy*' + pry* - r'y* + u(y*' -| py*): Lt' j* -l u,0 : r,Lr' - ,ly* - reíoo',r: I eIPd* rclx * c. Thus,!: uyngives (4). We shal1 see that this method extends to higher-order ODEs (Secs. 2.10 and 3.3). Problem Set 1.6, page 36 1. y' : 4, ' : -Il4, : -xl4 _l c* 3. ylx : c, y| lx : ylxz, y' : ylx,' : -*l,' + x2 -- c*, circles 5.2xy + r'y' :0,}' : -2ylx, ' : xl(2 ), '- x2l2: c*,hyperbolas 7. yg-rzlZ : c, !' : xj,' : -Il(x ),' : -llx,'l2 : -h |x| * c**, x : c*e-Ú'l',beIl-shaped curves (with x and interchanged) 9.y':-4xly,': l4x,4hl}l :h|.r| + c**,x-6*" 4,parabolas Il. xe-ala : c, j' : 4lx, ' : -rl4, : -x2l8 -l c* 13.Use dyldx: Il(dxlcty).(y - Zx)e": c,(y' -2+y -Zx)e" -- O, y' : 2- y * 2x, dxld - -2+' - 2xis linear, dxld * 2x : - 2, x : c*e-2u + yl2 - 5l4 15. u: C, Ll*dx -l urdy: 0,.}' : -LtrlL{r. Trajectories ' : uglw*. Now u : C*, u*dx * uody :0, y' : -u*luo. This agrees with the trajectory ODE in ul tf Ll*: ua (equal denominators) and ua: -u* (equal numerators). But these are just the Cauchy-Riemann equations. 17.2x -l Zyy' : O,!' - -xly. Trajectories}/ : lx,Inlil : tn|x| + c**, : c*x. 19. y' : -4xl9y. Trajectories ' : 9 l4x. : c*xgla (c* } 0). Sketch or graph these curves. Problem Set 1.7, page 41 1. In |x - xol { a; just takeb in x : blKlarge, namely, b: aK. 3. No. At a common point (xr, yr) they would both satisfy the "initial condition" y(xr) : lb violating uniqueness. 5. y' : í(x,y): r(x) - p(x)y; hence aflay : -p(x) is continuous and is thus bounded in the closed interval |x - xg| < a. 7. R has sides 2a and2b and center (1, 1) since y(1) : 1. In R, f : 2y' = 2(b + D2 : K, d : blK : bl(2(b + 1)'). daldb : 0 gives b : l, and dopt : blK : 1/8. Solution by dyly' : 2 dx, etc., y : ll(3 - 2x). l. |t + y'l= K : l + b2, u : blK, daldb : 0,b : 1, 0 : 1l2. Chapter l Review Questions and Problems, page 42 1I. dyl(yz + 1) : 4 dx,2 arctan}y : 4x 1- c*,! : ttan (2x + c) 13. Logistic ODE. y: llu,y' : -Ll'luz :4lu - Iluz,u: c*e-a* * i, 15.dyl(y2 + 1) : x2 dx, arctan y: xsB + c,y: tan (x3l3 + c) 17. Bernoulli, y' l xy: xly,u: !2,u' :2yy' :2x - 2xu linear, Ll : e-*' (In"' 2x dx * c) : I l ce-*', j : XC,. O,write yy' : -x(y2 - 1) and separate. A7 App. 2 Answers to Odd-Numbered Problems 19. Linear,J : e"o'*(Íe-"o." sin xdx -| c) : ce.o"" + 1. Or by separation. 21, Not exact. IJse Theorem 1, Sec. I.4; R:2lx, F : x2;the resulting exact oDE is 3;r2 sin 2y dx + 2x3 cos 2y dy : d(x3 sin 21), x3 ,i, iy: c. Or by separation, cot2y dy : -3l(2x) dx, etc., sin 2y : cx-r, 23.Exact. ll : I M dx: sin xy - x2 + k, ua :Jcosx} + k' : N, k : yr, sinry - x2 + y2: c. 25. Notexact. R* : 1 in Theorem2, Sec. 1 .4, F* : eU.Exactis eu sin (y - x) dx -| eafcos (y -.r) - sin (l,- x)] cly:0. ,,,: I M dx : ea cos(y - x) -ť k, u, : ea(cos(y - x) - sin (y - x)) + k' : IÝ, e'cos (y - xl : r. 27. Separation. y2 + x2 : 25 29. Separation. y : tan (r * c), c : -'ŽT 31. Exact. Ll: *'y' + cos,T + 2y : c, c : u(O, t) : 3 33. y' : xly. Trajectories '' - : c*/xby separation. Hyperbolas. 35. y : lo kt, e4k : 0.9, k : +ln0.9, ekt: 0.5, t: (In0.5)/k: (ln 0.5)/t(ln 0.9)/4l:26.3 [days] 37. ekt : 0.01, / : (ln 0.01)/k : IJ5 [days] 39. Y' : -4xly. Trajectories ! : crx'ln ot x : c2 a 41.LOgisticODE y' : Ay - By',y: l/u,u' + Au: *B,Lt: ce-At + BlA 43, A : amount of incident light. A thin layer of thickness Ax absorbs M : _kAA^x (-k : constant of proportionality). Thus MlLx : -kA. Let Ax -+ 0. Then A' : -kA, A : Aoe-k' : amount of light in a thick layer at depth r from the surface of incidence, Problem Set 2.1, page 52 1. y : 2.5e4* + 0.5e-4' 3. y : e-* cos x 7. Yes 13. No l9.ydzldy:4z,,y: (c l cr)-tl3 Problem Set 2.2, page 59 l.y: Cle7* * c2e-* 5.y: Cte0,9* l cre-l.t* 9.y: cle3,5* l cre-I.S* 13.y: clel2'+ c2e-l2* l7.y" - 2\/jy' + 3y:0 21.y:4e3'- 2e-* 25.y:2+e_o* 29. y : ,-o,1r13.2 cos 0.2x -| 1.6 sin 0.2x) 33. yr : g-t, y2 : 0.00le* + ,-* 5.y:4x2+7lx2 11. No 17.y: clekr + c2 3.}: (cr * crx)ez.s' 7. y : uo,sr(A cos 1 .5x -| B sin 1.5x) 11. y : Á cos 3rrx -| B sin 3rrx lÍ.y" - 3y' 1- 2y: g 19.y" - 16y:6 23. y : ,-2112 cos J - sin x) 27. y : (2 - 4x)g-o,zs* 31.y:4e5* - 4e-5* 9.Yesif a* 0 15. F(x, z, z') : g 21.(dz/dy)z: -z3 siny, -Ilz: -dxldy: cos j lčt,x- -siny * cly l c2 23. !"! : 2, ! : tG -| gstz - *, y(3) : Ť, y'(3) : 4 25.y": k!',r,t : kZ,Z: clek,: !',rr] 1,y: (ek* - I)lk 35.Write E - e-o'l2,r: cos ídí,s:sincox.Note thatE| : -žoE,c' : -0)s,S' : aC. Substitute,_dr9p E, collect c-terms, then s-terms, and use rr.,2 : 1r--"!or, to get c(b - *o' + io' - @1 + s(-ao + lao + !aai): 0 i 0 : 0. A8 App. 2 Answers to Odd-Numbered Problems Problem Set 2.3, page 61 1. 0, 0, -2 cos x 5. -I2x3 * 9x2 * 8x - 2, -28 sin 4x - 7.y: (cr i crx)e-2* 11.y: CI-3,7* l C2e-* 3. -0.8x3 l 6x2 + 0.4, O, eo,4* 4 cos 4x,0 9. l- : n-sr(A cos 2x * B sin 2x) 13. y : Á cos 4.2alx -l B sin 4.2ax Problem Set 2.4, page 68 l.y: yo cos agt l (uglag) sin r,ro/, Atinteger t (if ag: Ť), because of periodicity. 3. mLT" - -m8 sin 0 : -mgT (tangential component of W : *8),0" + ao20: O, @gl(2Ťr) : XQntln\. 5. No, because the frequency depends only on klm. 7. (í) Greater by a facto, XE. (ii) Lower 9.o*:Iao2 - c2l14*2)]a2: oo[1 - czl14mkllrl': ro(I- czl8mk):2.9583 11,Zrrla* since Eq. (10) and y' : 0 give tan (a*t - 6) : -ala*, tan is periodic with perioď rrla*, 13. Case (II) of (5) with ,: \/4*k: \á-O0-4500 : 3000 [kg/sec], where 500kg is the mass per wheel. 15.y: [yo -| (uo + ay)t]e-ot,!: [1 + (uo -| l)t]e-t; (ii) u6 - -2, -3l2, -4l3, -5l4, -615 I7. y :0 gives cl: -c2u-'F', which has one or no positive zero, depending on the \\-11 initial conditions.--..- .. ...-_. Problem Set 2.5, page 72 I. c63 + c2x-2 3. (cl * c2lnlr)*n 5. x|A cos (ln l"l) + B sin (tn lx|)] 7. cgl,a l c2x7,6 9. c7xo1 + c2xo,9 1L.3x2 - 2r3 13. x-o 5[2 cos (10 ln l"ll - sin (10 h lxl)] 15.2x-3 + 10 Problem Set 2.6, page 77 L.y" _ 0.25y:O,W - -1 3.y" _ 5. *'y" * 0,5xy' + 0.0625y : O,W - x-o,s 7. *'y" 9. ,'y" - 0.75y : O, W - -2 1l. y" 13. y" -l 2y' + I.64y : 0, W : 0.Be-2* 15, y" + 17. y" + 7.6rry' + 14.44 y : 0, 1ry - u-7,6rr 2ky' + k'y -- f "}' * 4y: 6.25y : 0, W 5y' + 6,34y : 0, W - u2kr 0,W:Zlx -r< O, W : 0.3e-5* Problem Set 2.7, page 83 I. cp-* + cre-2* + 2.5e2" 5. cpz* + cre-3* - x3 - 3x - 0.5 7. r-""(A cos 8x * B sin Bx) + e'(cos 4x + ! 9. cp-O,a* + c2e0,4" l 20xeo,4:r _ 2grr_o^r 1l. cl cos 1.2x,| c2 sin 1.2x -| 10x sin 1.2x !3. e-2"1A cos.x f B sin x) 4- 5x2 - 8x + 4.4 15. 4x sin 2x 3. cp4* + cre-4* sin 4x) - 1.6 cos 2x + 0.2 sin 2x + 2.4xe4* - 4e* App. 2 Answers to Odd-Numbered Problems A9 17. e-o,1r11.5 cos 0.5x - sin 0,5x) 12no,5r 19.2e-3" + 3e4'- I2x3 + 3r2 - 6,5x Problem Set 2.8, page 90 1. -0.4 cos 3r + 1 .2 sin 31 3. - 12.8 cos 5. 0.16 cos 2t + 0.t2 stn 2t 7. + cos 3r - 9. cp-tlz + c2e-3t'' - Ť cos / - f sin r 11. (c1 -l czt)g-sttz - $ cos 3t - sin 3t 13. e-t,st(A cos / f B sin t) + 4 + 0.8 cos 2t - 6.4 sin 15.0.32e- cos 5t + 0.68 cos 3t + 0.24 sin 3t I7.5e-at - 4e-2t - 0.3 cos 2t + 0.1 sin2t 19, e-',"(0.2 cos / - 1.1 sin /) + 0.8 cos t + 0.4 sin r Problem Set 2.9, page 97 |. LI' + R1 : E, 1 : (EIR) -l ce-RtlL - 2.4 + ce-5ot 3. RI' + IlC : O, I - ce-tl(Rc) 5. 1 : 5(cos / - cos l0t)l99 7,Io is maximum when,S:0, thus C: Il(azL). 9. R > Rcrit : 2\/LlC is Case I, etc. 11.0 13. cp-zo'+ ,re-lot + 16.5 sin 10r + 5.5 cos 10r í5.E' : -r-+t(7,605 cos * + 1.95 sin žt),t : e-o,1úlÁ cos lt + a sin }r) - e-4'cos }r 17. E(0): 600, 1'(0) : 600, I -- e-"(-l00 cos 4t + 75 sin 4t) + 100 cos 19. (b) R:2 d-L, L: 1 H, C : I|I2F, E: 4.4 sin 10r V Problem Set 2.10, page 101 1. Á cos x -l B sinx - J cos x * (sin x) ln |sin x| 3. clx * c2x2 - -r cos .r 5. (cos x)(ct -l sin x - ln |sec x i tan "l) + (sin x)(c2 - cos ;r) : (cr - ln |sec x -l tan x|) cos x l c2 sin x 7. (ct + Zň sin x -| (cz -l ln |cos x|) cos J 9. (ct * c2x)e" + x2 -l 4x + 6 - e'(|n l"l + 1) 11. c1 cos 2x * cz sin 2x + *" cosh 2x 13. clx -l c2x2 - x sin x 15. A cos,r -l B sin x l lpt + !pz, )pr zts }p in Example l, !p2: ťi sin 5x 17. u" + tl : 0 by substitution of j : ttx-'''. y, - x 1/2 cos x. jz - x-Ilz sin x, lp : -lattz cos -r + *r- 1/2 sin x from (2) with the ODE in standard form. Chapter 2 Review Questions and Problems, page 102 9, c-,e4* + cze-2* - 1.1 cos 6x - 0.3 sin 6x II. e-4"(Á cos 3x * B sin 3x) - t cos 3x * } sin 3x 4.5t + fr sin 3.6 sin 4.5t 3t 2t Al0 App. 2 Answers to Odd-Numbered Problems 13. y, : x3, j2 : x-4, f : x-5, W : -Jx-', jp : -b.r-' - }x-3 : -á"-' 15. yr : gI, !2: x ', W : e2*, lp: e"l(2x) 17. yr: e* cos x, !2: e* sin x, W : e2*, !-p : _ xe'cos.r i e'(sin x) ln |sin x| 19.y:4e2* + 2e-7* Zl.y:9x-4 + 6x6 23.y - e-2* -2e-3* i 18x2 - 30x + 19 25.y: *"'+ 4x2 - 5x-2 27.y - -16 cos 2t + 12 sin2t-l 16(cos 0.5r - sin 1.5r). Resonance for al(2rr) : 2l(2t) : Ilrr 29. a: 3.1 is close to rrg : \/H*: 3.) : 25(cos 3r - cos 3.It), 31.R:9 C), L:0.5H,C:0.025 F,E:17 sin 1tY,hence 0.5I" + 9I' + 40I : IO2 cos 6/, I - -8.I6e-Bt + J.Se-rot + 0.66 cos 6/ + I.62 sin 6r 33. E' :220.3I4 cos 3I4t, I - u-sot(A cos 150r f B sin 150/) + 0.847001 sin 314r - I.9852I9 cos 3I4t Problem Set 3.1, page 11'| 7 . Linearly independent 11. x|x| : x2 if x ) 0, linearly dependent 13. Linearly independent 17 . Línearly independent Problem Set 3.2, page 115 I. y"' - 6y" -l llyl - 6y : O S.yu" l4y" : g 9. cle* -| (cz l cgx)e-" 9. Linearly dependent 15. Linearly independent 1,9. Linearly dependent 3.yu" - } : 0 7. c, -| c2 cos x -l ca sin x lI. cp' + c2eQ* )* + czel-.'/i>* 13. ro,25r -l 4.3e-o,7* + I2.1 cos 0.1x - 0.6 sin 0.1x t5.2.4 * ,-t,6r(cos 1.5x - 2 sin 1.5x) 17. y : cosh 5x _ cos 4x t9.y: clx-2 l c2x + cgxz.W: I2lx2 Problem Set 3.3, page 122 1. (cr + crx)ez* + cge-z' - 0.04e-3* + x2 * x * 1 3. c1 cos lx -l c2 sin }x -l x(cg cos }x i cn sin Lň - le-" sinlx 5. clxo,s l c2x + cgxl1 + 0.1-15,5 7. c1 cos x l c2 sinx * c3 cos 3x * ca sin 3x -| 0.2cosh2x 9. y : (4 - x2)e3" - 0.5 cos 3x + 0.5 sin 3x ll.x-z - x2 + 5xa + x(lnx + 1) 13. 3 + 9e-2" cos 9x - (1.6 - 1.5x)e" Chapter 3 Review Questions and Problems, page 122 7. c1 l czx'l' l crx-llz 9, cp-o,s* + C2eo,5* + Cre-L,s* 11. crxz(lln x - fr) + crxz l cgx -l ca -l |x7 13. cp-* + e"lz(c2cos (}V5x) * ca sin (}\6x1) + 8e"l2 15. (c1 l c2x)e* l cge-* l 0.25x2e" 19. cos 7x 4- e3* - 0.02 cosh x 17. -0.5x-1 + 1.5x-5 App. 2 Answers to Odd-Numbered Problems Problem Set 4.1, page 135 1. Yes 5. yi : 0.02(-y1 -| yz), yL : 0.0Z(yr - 2y, 1- ys), y! : 0.02(y2 - }s) 7. rr: l, c2: -5 9.3 and 0 11. Y', : !2, YL : 4yt, lt : ct 2t + c2e2' : ,, jz : !'t 13.Yt: !2,yL: yz, eigenvalues 0, 1, jt : cl l c2et,yr: y'r: y' 15. Yi : !2, y' : 0.7O9375y1 + 0.75y2 (divide by 64), h : cte-,o'.I2st l crgo.tlst Problem Set 4.3, page 146 1. Yl : cte-6t l c2e6t, ;1 z: -2cru-u' + 2c2e6t 3. Yr : Cle2t t ,r, jz: Ct 2' - ,, 5. yr : cre4i' + c2e-ait : (ct t ,r) cos 4t 4- i(c1 - cz) sin 4t : Á cos 4t + B sin 4t, y2 : icp4Žt - icze-4it : (ict - irr)cos 4r -| i(ic1 4- ic) sin 4t : B cos 4t - A sin 4t, A : r, l rr, B:i(c1 -cz) 7.Yr:2c, * c2 -6t,lz: -ct l cge-6t,!3: -ct - 2(c2 + cg)e-6t 9. Yt: cl,eL'a' l 2c2e-o9t^+ 2cge-7Bt, y2: 2cpl,Bť + c2e-o.9t-_ Zc"e-t.at, y3 : 2cpl.at - 2cre-o.St + cge-L.Bt 1I. yt: l0 + 6e2t, lz : -5 + 3e2t 13. y, : 2.4e-t - 2r',", lz : I.8e-t + 2e2.5t 15. yr - 2nla,ft + 10, !z: 5ela," - 4 17. yr : yI + h, lL : y'| 1- y',, : -!t - lz : - rt - O',i yr), yi + 2yi + 2yt : O, yL : e-t(A cos / * B sin_r), !2 : y| + yt : ,"(B.o, r - a .i" /). Note that 12 : yt2 * yz2 - r-ztl{z + B\. 19. It: 4cle-2oot + cre-'O',Ir: - c. -2OO _ 4c2e-5ot Problem Set 4.4, page 'l50 1. Saddle point, unstable, lt: c' -4t + cre4t,!2: -2cp-at + 2c2eat 3. Unstable node, lt: ct t + c2e3t, ;1.z: -c.s,et + cre3t 5. Stable and attractive node, lt: ct 3t + c2e-5r, j2: cI -3t _ cz -5t 7. Centet, stable, lt : A cos 4t i B sin 4t, yz : -2B cos 4t + 2A sin 4r 9. Saddle point, unstable, lt : ct 3t + c2e-t, !2 : cte3t - cz -t 11. yr : y : clekt + c2e-kt,!z: !', hyperbolas k2y] - yz2 : const 13. y : n-2t14 cos / i B sin r), stable and attractive spirals 17. For instance, (a) -2, (b) -1, (c) -*,(a) 1, (e) 4. Problem Set 4.5, page 158 1. (9, 0):/i : !2, YL:3y1, saddle point; (0, -1), }r : t, !2: -I t z, 'r: - r, ,: 3 r center 3. (0, o), ri_: 4yr,yL: 2!t, saddle point; (2,0),lt: 2l ,,lz: z, 'r: 4 r, ),z: -2h,center 5. (0, 0), yí: -jt l yz, yL : _yt - !z, stable and attractive spiral point; (- 2,2), !t:-2t t,!z:2* ,,|:- t_ 3r, l:- t- ir,saddlepoint All App. 2 Answers to Odd-Numbered Problems 7.y'r: yz.yL: -}r(1 - 4yr), (0,0), y't: yr,yL: -yt, center; (á, 0), lt: * * rr, lz: z, 'r,: ,, yL: r-i - )(- ), L: }r, saddle g. (*, + Znrr,0) saddle points; (-i, + 2nr,0) centers. Use -cos (ti, + }r) : sin (tir) - t!r. ILy'r: !2,yL: -yr(2 + y)(2 - )r).(0,0), yL: -4y-,, center; (-2,0), L:8 r, saddle point; (2, O), L : 8 , saddle point I3.y"ly' + 2y'ly:0, ln y' + 2ln}: c,y'y2: !z!t2: const 15. y : Á cos t + B sin /, radius Problem Set 4.6, page 162 \/A\ B' 3. yr, : Á cos 4t + Bsin 4r + #,!z -- Bcos 4/ - Á sin at - f;t 5. yr : cte4t + c2e-3t l 4, yz : cI 4t - 2.5c2e-3 - 10 7.yl,:2cp-9t * c2e-4t - 90t + 28,!z: ct 9t + cre-4t - I26t 9.yt : cpt l 4c2e2t - 3t - 4 - 2e-t,jz : -ctet - Screzt + 5t + 11.y, : 3 cos 2t - sin2t + t l l,yz: cos 2t + 3 sin2t + Zt - l 13. y, : 4e-t - 4et l u",!z: -4e-t + t 15. y, - 7 - 2e2t + e3t - 4n-"', lz : -e2t 1 3r-3t 17.I| + 2.5(It - Ir): 845 sin /, 2.5(I; - Ij l 2512: 0, 1, : (95 + I62.5t)e-5t - 95 cos t + 312.5 stn t, Iz : (-30 - I62.5t)e-5t + 30 cos t -l I2.5 sin t 19. I| + 2(IL - Ir) : 2O0, 2(Iz - 1r) + 8I2 + 2 I Iz dt : 0. 11 : 2cp^,t + Zcze^, + 100, Iz: (7.I + ÝMl)crgx,t + (1.r - \/Ml)rrun,', it : -0.9 Xz: -0.9 - \/Ml Chapter 4 Review Questions and Problems, page 163 + \/MI, 11. y, : c7 Bt * c2e-Bt, yr:2cpgt - 2cre-B'. Saddle point t3. y,: c7 t l c2e-6t, lz: ct t - 6c2e-6. Saddle point 15. y, : ctez,St l c2e-3t, lz: -cle1,,t + 0.75c2e-3ú. Saddle point 17. y-,, : c7 5t l c2et, lz : ct 5' - ,rr'.Unstable node 19. y, : e-'(A cos 2t * B sin 2t), y2 : e-t(B cos 2t - A sin 2t). Stable and attractive spiral point 21.yr: clt l c2e-t + e2t l u-", lz: -cze-t - I.5e-2t 23.yr: clt + c2e-2t - 6e-t - 5, jz: -cte' - Zcre-2t + IOe-t + 6 25.yr: cI 3t + c2e-t + - 2t +2, jz: ct3'- rrr-'- t2 + 2t - 2 27. A saddle point at (0, 0) 29. h - 4r-au - e-Iot, Iz: -e-4ot + 4r_1,ot 31. (nrr,0) center for even n and saddle point for odd n 33. Saddle points at (0,0) and (Ž,*), centers at (0, }) and (á,0) Problem Set 5.1, page 170 t. ag(I * x * **' +,,,) : aoe* 3. ag(I - 2x2 + ?*n - + . . .) + a{x - 3r" + Zrr' - + . . .) : ag cos 2x * !a1 sin 2x +14 7.5 -l e-t App.2 Answers to Odd-Numbered Problems Al3 1,c:aolI 3. 2 (as function of / : (x - 3)'). Ans. \/Ž 7.2 ll. n t5.i t' - 1)'..ť": R : z -- (s - 3)!s:D 5. ag(I + i*) 7. oo l agx + (*ao + *)r'+ . . . : ao r 1- e* - x - I : ce* - x -9, oo l alx + lag2 + . . . - ag - a1 l ale* 11. s : Z - +, + 8x2 - Šr"+ Ťrn - Ť**r, s(0.2) : 0.69900 13. s : L + i* - Lr*" + 4*L"u, s(1) : 0.73125 15. s : I 4 x - x2 - 3"' + lrn + *ď', s(+) : 9á8 Problem Set 5.2, page 176 1. 1.1 5.0 9. l 13.; l, ""^1 .r"; R: I ^ J(.l - Z)s:J 1.y1:1 * 3,Yr: 1 sinh r...: X l7.ao(I -Lrrn --L*5 -...) 1- a{x+*r' +á"' + h*n - h*u -...) 19.ag,l ar(x-3*" +?ru -#r'+&*n #rr'+ -...) 2l. aoQ - t*' - *nrn + &*u + . . .) * a{x - á"' - *r'+ mfu"' + . . .) 23.ag(I + x2 + x3 + xa + x5 + x6 + ...) * ap Problem Set 5.3, page 180 3. P6@) : #(zt7x6 - 3l5xa * l05x2 - 5), Pz@): rtqz9x7 - 6g3x5 t 375x3 - 35x) 7. Set x : az. ! : ctPn@la) + c2Qn(xla) 15. P11 :1/1 - *z, Pr' : gr\/t - ,', pr' :3(I - x2), Pn' (t - x2xtO5x2 - 15)/2 Problem Set 5.4, page 187 1 x x3 coshx t,---T---r--r... x2!4!x lz: 9!t| I44 36 x2 25xa nx- ť - ť*r- rcu* 5.r(r-1)+ 4r-l 2:0, l.1 --1,f2:- 2;yt:1 -:**-+ ., x6I20 x2 x4 --r 24 120 7. Euler-Cauchy equation with r : x l 3, lt : (x * 3)5, lz : ltln (x + 3) 9. bo: 1, Co : 0, 12 : 0, yt : g-r,lz : e-í Inx 11. yt : I/(x + 1), yr: l/x 13. bo : l, ro: O, ťl: l, ,r: 0, yt : ,rlz7l * 2x + 2x2 + **'+ . . .), jz:I-1 2xl2x2+", 15.yl : (x - 4)7,yz: (x - 41-5 (Euler-Cauchy withr : x - 4) 17.yr:J* x3 -irn+ t"'- **u +. .,!z:1+ 3x2-á"r+ |rn_É*'r 11__6 l -r 56"{ --r . . . ,X' -+3! x' 12 11Vo - ---a- x"2 A14 App.2 Answers to Odd-Numbered Problems 19. y : ctF(ž, i,ž; ,) + crÝiFlt, l, }; x) 21.y: Á(1 - 4x + 3"') + B\,'vF(-ž,Z,Ž; ,) 23. y : clF(2, -2, -i t - 2) -l cr(t - 2)"l'F(Z, -+. }: t - 2) Problem Set 5.5, page 197 1. Use (7b) in Sec. 5.2. 3. 0.] 1 9 58 (exact 0.1 6520), 0.1961 4 (0.22389), - 0.2] 651 ( - 0.26005), -0.39788 (-0.391I5), -0.17038 (-0.I]160), 0,15680 (0.15065), 0.30086 (0.30008), 0. 16833 (0. 17165) 5. y : c{,(Ax) * c2J_,(Ax), u * 0, -|1,,,, 7.y : crJ,(Ýi) -l c2J_,(Ýi), u * 0, *1,. . . 9. y : cgJ1(2x), Jt J_t linearly dependent 11. y - x "[c]Jx) -| c2J_,(x)l, u * 0, *1, 13. y : clJ,(xs) * c2J_,(r'),, + 0, *1, . . . 15. y : ct iJ{Z\G), Jt, J_1,linearly dependent 17. y : *'lnJr(3,r'ln), Jt, J_t linearly dependent 19. y : x2l'(ctJa,u{,4x7l4) l c2J _g15(4r'ln)) 21. Use (24b) with z : 0, (24a) with u : I, (24ď) with u : 2, respectively. 23. J-(x) : Jn(xz): 0 implias xl-nJn(xr) : x2-nJn(x2) : 0 and|x-nJn(x)]' : 0 somewhere between x1 and xzby Rolle's theorem. Now use (24b) to get Jn*r(x) : 0 there. Conversely, Jn*,(xs) : Jn+íxs) : 0, thus ,r'*'Jn*,(xs) : x4n*lJr*7(x+) :0 implies J.(x): 0 in between by Rolle's theorem and (24a) with u : n l I. 25.Integrate the formulas in (24). 27.IJse (24a) with z : l, partial integration, (24b) with v: 0, partial integration. 33. CAS Experiment. (b) ío: 1 , xl : 2.5, x2 : 20, approximately. It increases wtth n. (c) (1a) is exact. (d) It oscillates. (e) Formula (24b) with z : 0 Problem Set 5.6, page 202 l.y: c{5(x)*c2Y5@) 3.y: crto],,G;+crYo(\G) 5.y: crJr(x2) l c2Y2(x2) 7.y: x-s(crJr(x) + c2Y5@)) 9. y : x3(crl"(x3) * czIr("')) 11. Set g{D - kHQ), use (10). 13. Set x : is in (1), Sec. 5.5, to get the present ODE (I2) ínterms of s. Use (20), Sec. 5.5. Problem Set 5.7, page 209 3.Setx:ctlk. 5.x:cosg,dx- -sinOd1,etc. 7. 1rr: (mrrl5)2, ffi : I,2, " , ; lrr: sin (mnxlí) 9. Lrr: [(2m + l)rrllL]',*:0, 1,, ,, y*(x): sin |(2m + I)rrx/ZL] 1I. ^n - *',ffi: O,l, , , , i }o : I,!rn: cos mx,sinmx,m: 1,2, -. 13.k: k-from tank- -k. lr,-: k*2,m: I,2,",; !.,n: sinkrnx 15. ^rn: m2,ffi: I,2, " , i !m: x sin (mln|x|) 17.p -- eB*,q: O,r: eB', h*: m2;yrn: g-4r sinmx,ffi:1,2,... t9. ^-: (mrr)2, lrn: _r cos mŤrx, x sin mŤrx) ffi : 0,1, . . App. 2 Answers to Odd-Numbered Problems Problem Set 5.8, page 216 1, 1,6Pa@) - 0.6Pg(x) s.}r"lx; - lPr@) + $rrlx) - !Po@)7. -0.4]75P1@) - 0.6908Ps(x) + 1.844Pr(x) - 0.8234P7@) + ď:s++Pg(x) + . . . lTlo: 9. Rounding Seems to have considerable influence in probs. 6_15. 9,03799Pz@)+ L673Pa@) - 1.397P6@) +0.3968PB(r) + -.,l7lo:8 11. 1.175Pg(x) + 1.I04Pr(x) + 0.3575P2@) + 0.0700P3(x) ,ffio:3 or 4 13.0.7855Po@) - 0.3550Pz@)+ 0.0900Pa(r) -. ..,frlo:4 15. 0.I212P6@) - 0.7955Pz@) + 0.9600Pa@) - 0.3360P6(x) + . . . , tfro: 8 17 . (c) a* : (2 l J 12 ( ao,ň)(J {ao,,,) l oq*) : 2l (ag,o,J {ao.,")) Chapter 5 Review Questions and Problems, page 217 11. e3*, e-3*, or cosh 3x, sinh 3x 13. e*, 7 -| x4 19. 0 if t < 2, (t - D424 if t > 2 23. e-t sin t 31. cosh XEt - t l 13. (e-2'r2t _ u-4s, 4, s S-Ťr 17. sin t if 2rr < t < 8r'r, 0 elsewhere 2l. u(t - 3) cosh (2t - 6) 25. e-2t cos 3r * 9 cos 2t + 8 sin 2t l ,, t 29. ," (cn'r - l) - -k,k 1) -^ (; - :) ,-'" - (; - 1n) ,-,* 27.sin 3r + sin rif 0 < t < rrandf sin 3r if t} rr 29. t - sin r if 0 < t { 1, cos (t - I) i sin (t - I) - sin t if t > l 3l. et - sin t * u(t - 2rr)(sint - lsin}t) 33.t: I +7, " + 4 :8(1 + Ď'(t - u(T- 4)),cos 2t+2t2 - 1if t15, cos 2t + 49 cos (2t - 10) + 10 sin (2t - 10) if t > 5 35.Rq' + qlC: O,Q:9(q),q(0): CVo,i: q'(t),R(sQ- CV * QlC:0, q : CVge-tl(RC) l00 l00 o- - e-2, (! - ' )., : 0 if r l2 and37.1u *;I: ,z e-''.l: \s s+l0/ l -l0tt -2l|-( íít>2 App. 2 Answers to Odd-Numbered Problems Al7 39.i: e-2ot + 2Ot - 1 + u(t _ 2)|_20t + 1 + 3ge-2oft-2)l 41.0.7i' + 25i: 49Oe-5t|1 - u(t - 1)], i:20(e-5t - e-25o1 + 2Ou(t - 1)7-u-st t ^-25ot +245l-Ť e -"" -,"| 43. i : (10 sin l0r + 100 sin t)(u(t - rr) - u(t - 3n)) 45.i' +2i+zf^rrodr:1- u(t_ 2), I: (| - e-2')/(s2 +2s +2), i : e-t rln , 1 u(t - 2) g-t+z sin (t - 2) 47. i:2J cos t + 6 sin r - e-t127 cos 3r + 11 sin 3r) -| u(t - 2rr) 1-2] cos / - 6 sin 7 a n-ft-2-)(27 cos 3t + 11 sin 3r)] Problem Set 6.4, page 247 1.y: 10cosrif 0 2rr 3. y : 5.5et -l 4.5e-t * 51ut-tt2 - e-t+I'1u(t - ) - SO(et-l - r*t+I;u(r - 1) 5. .y : O.I|et + e-2'(-cos t + ] sin r)] + O,Iu(t - 10X-et + e-2t+3o1cos (r - 10) - 7 sin (/ - 10))] 7. y -^l + te-t sin 3r -| u(t - 4)l- 1 a r-t+a(cos (3r - 12) + j sin (3t - l2))l - *"G - 5)g-t*s sin (3r - 15) 9.Y : 5t - 2 - 50u(t - lr)g-t+'stn2t. Straight line, sharply deformed between zr and about 8 11.y : (0.4t + I.52)et + 0.48e-4'+ L6u|t - 2)l-e, + e-4t+Iol 19. Y : 3/((s2 + 4)(s2 + 9)), y : 0.3 sin 2t - 0.2sin 3r 21. (s2 + 9)Y : 4 + *_í' * e-^)/(s2 + 1),y : sin / f sin 3t if t 1 r,tsin 3r if t ) n l 23.}if 0 < t < 1.Z Jrsin (2(r - 1)) dr : -Zcos (2t - 2) + f;it t > t 25.y - 2e-2t - e-4t l (g-zt+z - e-4t+4)u(t - 1) + (e-2t+4 - e-ar*Blult - 21 Problem Set 6.5, page 253 1.t 1 5. - stn ot 9. *@"' - ,-u,) 13.r-sinr 27.y -l*y:l,!: t 3l. Y(I + l/s2) : I/s, y : cos / Problem Set 6.6, page 257 4 1._(s - 1)' 2s-l 4 3.et-t-1 ,. + (elt - ,-,o') : +sinh frr í.+G - } cos 2t): lsinz t 15. $(cosh 3t - I) 29.y - y* sin / : cos /, Y : lls,y : 7 33. Y(1 + 2/(s - l)) : (s - l)-2, y : sinh / 2as TF +;FF 24s2 + 128 lp-Gtr 2s cos k -l (s2 - 1) sin fr 1s2 + t12 3. 5. 7. 11. (r'+4s*5)2 A18 App.2 Answers to Odd-Numbered Problems Problem Set 6.7, page 262 1. yr : -e-t sin /, lz: e-t cos / 3. y, : 2e-at - 4r", lz: e-4t - 8e2t 5. yr, : 2e-t + 4e-2t + Lt - i, lz : -3e-t - 4e-2' - i, + i 7.yt: e-t(2cos2t -l 6 sin 2t) + t2, y2: IOe- sin 2t - t2 9.yr:4cos 5t + 6 sin5r - 2cos t - 25sínt, lz:2 cos5r - 10sin5r i 20sin/ ll.yr: -cos / + sin t + I -l u(t - 1)t-1 * cos (t - I) - sin (/ - 1)] }2 : cos r * sin t -l -| u(t - 1)t1 - cos (r - 1) - sin (r - 1)] 13. y, : 2u(t - 2)(rn' - e'*6), !z: e2t -l u(t - 2)(en' - 3r2t+a + 2et*6) 15. yr - -e-2t + et + á"(t - I)(-g-zt*" + ,'), jz: -e-2t + 4et + luQ - I)(-e-2t+3 + et) I7.yr: 3 sin 2t l 8e-3t, lz: -3 sin2t + 5e-3t 19. y, : e' - e-', !2: t, j3: e-t 25.4il + 8(i1 - i) + 2i|:390cos/, 8i2+ 8(iz - ir) + 4i!2:0, L: -26e-2t - 76e-8t + 42 cos / + 15 sin /, iz: -26e-2t + 8e-Bt + 18 cos / -l t2sínt Chapter 6 Review Questions and Problems, page 267 13.6te-t 17. t2ekt l 11. ^ (s - 3)' 'l1 (s-l)(s"+4) 23. IO cos t\/ž 29. te-zt Sin t ,+ 15. íe-' stn 1 19. ln s - ln (s - 1); (et - I)lt 2 13. ^ s(s'* 4) 2s2 19. ^ s"-1 25.3e-2t sin 4t 31. (t2 - I)u(t - I) a-b (s-aXs-b) 27. u(t - 2)(5 + 4(t - 2)) Ťr '3. é (ot - sin ror) 15.e-^ (+ - +) 21. 35.20 sin r -| u(t - 1)t1 - cos (r - 1)] 37. I0 cos 2t - } sin 2t + 4u(t - 5) sin (2t - I0) 39. e-t17 cos 3r -l 2 sin 3t') 4I. e-t -| u(t - ŤĎrL2 cos t - 3.6 sin r -| 2r-t+r - 0.8 e2t-27rl 43. u(t - 1Xr - I)gzt-z + 4w(t - 2)(2 - t)g2t-+ 45.yr: e'+žr-'- }cos t -lsin/, lz: -et +*r-' +}cosr + jsinr 47. y, : i,r-' sin 2t, y2 : e-t(cos 2t - } sín2t) 49. yr: e2t, lz: ezt + et 51. 1:(1 -e-2";/;sls+ 10)], i:0.1(1 - r-tot\ +O.Iu(t-2)1-7 + e-lot+2ol 53. 1 : e-2'(76cos4t - 42 sin4r) - 16 cos 20r + 16 sin 20r 55.r1 + 10(i1 - i): I00t2, 30t!r+ rc(i;- i'r)+ 100i2:0, i: (t + 4tle-5' + IO - !, iz: (t + 2t)e-5t + zt - t Problem Set 7.1, page 277 -36 6l 36 -18 l 54 o_] ,:,:^ {] ,|:,:] Same, 3Undef|;:][-ij] [ -4B -21 |- 36 0 48 ,. l 38 -oo| ,l -r, 24 24 L 67 - ,r.j L ]2 ;. -48 ,[ :] [ i;] ame [,ii] 9. -5*, - -3 -5x, -| 2x2: 4 -3r, l 4x2: 0 Problem Set 7.2, page 286 ,|1:1undef ,|:l]|i,li:T,] ,|:,:i :,,]|,,1:^ :|:] [ |,:,: :: ;] r565l 34 ,,,l ,rrl . L,no_] [ -:to n0 | -u, 34 |-rro 68 |- 25,7 , same, | 'r' l,- ror undef. App. 2 Answers to Odd-Numbered 5. |20 -3 -7l, |-62 :1] , Same, [-]i -5.0 1.8 3.5 44 64 64 110 72 -II4 Same ',,';1] Same 10l ,l , o) ?,'I] [ts 0 401 ,,| 3 0 ,|,r,, L6 0 ,u_] ' t 331 8 -160l ,. | 252 49 -u* | L -gos 52 zn ] Al9 A20 App. 2 Answers to Odd-Numbered Problems |- 324 32 I 1l. l 244 38 I L-z++ - l0 Problem Set 7.3, page 295 1.x:2.5,y:-4.2 5.x:O,y: -2,Z:9 9. x : 3y + 2,y arb.,, : -y,| 6 13.w : I, ! : 2Z - x, X, 7 aíb. Problem Set7.7, page 314 5. 107 9. -66.88 13. u3 + u3 + w3 - 3uuw 19. x - -1.2,y : 0.8, z : 3.1 23.3 |- 7060 960 I . l 1548 1246 I L-8l40 - l090 3.x:0.2,y:1.6 7.x:4.y:O.z:-2 11.y:2x+3z*I,x,zarb. t5.w:3,x:0,}--2,Z:8 7. cos (a + F) 11. 0 15.4 21. I :il] ";l]:,1:] L lil ,,,:^: |- 4324 I52o -4816l lll 3636 1242 -4518 l llL -3700 - |046 5002J 13. 83, 166,593,0 19. (d) AB : (AB)-: BTAT: BA, etc. (e) Ans. If AB : -BA. 2t.Tríangular are Ur t U2, UlU2,Ur', Lr t L2,LIL,,LI2. 23. t0.8 I.2]-, |0.76 I.24]r, p.752 I.248lT 27.p: [110 45 80]T, v : [92000 86300]T 17. II: (Rr + R)Eol(RrRr), 12: EglRb Ig : EglR, [Amps] 19. II - 12 - Iz : 0, (3 + 2 + 5)11 + I0I2 : 95 + 35, I0I2 - 51s : 35, 11 : 8, Iz:5, Is:3Amps 2l. x1 * x4 : 500, x1 l x2 : 800, x2 f x3 : 1100, xg * xa: 800, x1 : 500 - x4, x2: 300 l x4, x3 : 800 - x4, xaarbitrary Problem Set7.4, page 301 1. l, [1 -2]; ll 0 -3]T 3.3,[1 4 0 ]],l0 -2 1 3],[0 0 5 105]; t-2 4 5]T,[0 1 5]-, t0 0 1]' 5.2, |3 0 5], [0 3 al 3 0 5]T, [0 3 4]' 7.2,|8 0 4], [0 2 0], t8 0 4 0]-, [0 2 0 4]' 9.3, |I 0 3 0], [0 5 8 -37], [0 0 -14 296]; same transposed 1I.4,|I 0 0 0],[0 1 0 0],[0 0 1 0],[0 0 0 1]; sametransposed 13. No 15. No 17. Yes 19. Yes 21. (c) l 27. 2, |I - 1 0], [0 0 1] 29. No 31. 1 , |-+ * 1] 33. No 35. 1,[5 z 3 z 1] App. 2 Answers to Odd-Numbered Problems A2l Problem Set 7.8, page 322 |- 1.80 -2.32f l.| l s. L-O.zs 0.60_] f cos 20 --sin 20f l' Lrin ze cos 2ť" | '' | '?J Lo li] "|:',o |] Problem Set 7.9, page 329 1. yes, 2, [3 5 0l-, [2 0 -5]' 3. No 5.Yes,2,[0 0 0 1 0]T,[0 0 0 0 l]|- 0 ll 7. Yes, l, | | l. No L-t o_] 11. Yes, 2, xe-*, e-* 13. [1 0]T, [0 1]T; [1 1]-, [-t t]T; [t 0]r, [0 -1]T 15. x1 : -0.6yt -| 0.4y2 17. x1 : 2y, * lz Xz: -0.8y, -F 0.2y, xz : 5!t 1- 3yz 19. x1: 5y, -| 3yz - 3ys xz: 3!t -ť 2y, - 2ys xs : 2yt - yz f- 2ys 11. No inverse 23. t6\,E 29. 4u, - 3ur: 0, v : t [8 á]' 13. x : y i 6, z : y, y arbitrary l7.x:'l,y - -3 9.A-1 :A | 4 -| -5l 15. (A2)-1 : (d-r;z : l ts l _5 l L5 4 ,] 19. AA-1 : I, (AA-')-' : (A-t;-to-1 - I. Multiply by A from the right. 21.detA - -1-Cn: Czt: Cs3 - -1, the other C3paíeZaro. 23. det A : 1. Ctt: l, Cn - -2, Czz: 7, Cs: 3, Czz - - 4, Css : 1 2t. \/i6 )<) Chapter 7 Review Questions and Problems, page 33O 1l.x:4,y:J 15.x:i,y:-+,z:ž 19. x : 2z, ! : 4, z arbitrary 23.638,0,0 27. 14, ,^|i: ; ,:] 21. 0 |- 8.0 -3.6 I 25.| -s,e 2.6 I L 1.2 2.4 29. L-20 9 -3], ):| ["| A22 App. 2 Answers to Odd-Numbered Problems 3l. 2,2 35.2,2 39. 33.2, 37.# 4í.# 2 t;;] |',,';'::;l] 5l -,l ^)11 A, 13 18 A, 13 -5 10 11 5-[ ""L It: It: 43. 45. 23 -10 33 A, 12: 12A, 12: :22A :6A Problem Set 8.1, page 338 t. -2, |I 0]T; 0.4, [0 1]' 3.4,2x1 + e4 - 4)xz: 0, say, x7: 4, xz: l; -4,10 1]' 5. -4,|2 9]T; 3, [1 1]- 7.0.8 + 0.6i, [1 -il- 0.8 - 0.6i, [1 i]- 9. 5, [1 2]T;0, |-2 1]T l|.4, II 0 0]T; 0, [0 1 0]'; -1, [0 0 1]- 13. -(^3 - 18^2 +99^- 162)/(^ - 3): -(^' - 15^ + 54);3,|2 -2 1]T; 6, [1 2 2]T;9, L2 I -2]- 15. 1, [-3 2 10]T; 4, [0 l 2]T;2, |O 0 1]T 17.-(^3 -]^2 -5^ +15)l(^ +3): -(^' - 10^ +25); -3,[1 2 -1]-; 5, [3 0 Il', 1-2 1 0]T 19. -()" - 9)3;9, |2 -2 1]T; defect 2 21.^(^3 - 8^2 - rc^+ I28)l(^- 4)-- ^(^'- 4^-32);4,|-l 3 I 1]-; -4,1I 1 -1 -1]T; 0, [1 1 1 1]T; 8, [1 -3 1 -3]- 23.2, 18 8 -16 1]T; 1,[0 7 0 4]T; 3,[0 0 9 27T,-6,L0 0 0 1]- 25.(^+D2(^2+2t-15); -1,[1 0 0 0]',[0 1 0 0]-; -5, [-3 -3 1 1]T, 3, [3 -3 1 -1]- 29. Use that real entries imply real coefficients of the characteristic polynomial. Problem Set 8.2, page 343 ,[-; (-x, 0 0l l- 1l r0l ,_] , -r, Lo_] , '. Lr_] ,unr point (x,0) on thex_axis is mapped onto ), so that [1 0]T is an eigenvector coffesponding to .tr : -1. I- 1 0l T 11 1-0l 3. (x. y) maps onto (x. o' Lo o_] , ' L.] ; 0. L,_] o point on the x-axis maps onto itself, a point on the y-axis maps onto the origin. 5. (x, y) maps onto (5x, 5y).2 X 2 diagonal matrix with entries 5. 7. -2, Ll -1]T, -45,; 8, [1 llr,45.. 9. 2, |3 - 1]-, -I8.4",7, II 3lT, ]I.6" 11. 1, |-tt..',/s If, 112.2"; 8, [t U\,G],22.2" 13. 1, [1 llT,45"; -5, [1 -1]-, -45" 15. c[15 24 50]T, c ) 0 l7. x: (I - A)-'y : [0.73 0.59 1.04]T (rounded) 19. tl 1 1]T 2l. I.8 23.2.1 App. 2 Answers to Odd-Numbered Problems Problem Set 8.3, page 348 3. No 7. No since det A : det (Ar) : 9. Orthogonal, 0.96 -+ 0.28i 13. Symmetric, 9, 18, 18 17. Symmetric, a l 2b, a - b, a Problem Set 8.4, page 355 5. A-1 : 1-4r;-1 - _(A-t;r det (-A) : (- 1)3 det A : -det A : 0. 11. Neitheí,2,2, defect 1 1,5. Orthogonal, I, i, -i -b 1.t1 2]T,[2 -1]-; - : [] _'r1," : [; ;] 3. t| -l]T. [l ll-, D : [; :] 5. t2 -1l', 12 l]T, diag (-2, 4) 7. LI 0 0]T, [1 -2 1]T, [0 1 0]-, díag (I, 2, 3) 9. t0 3 2]T, [5 3 0]T, [1 0 2]T, diag (45,9, -2]) 13 [16 -,f ,-, []] ,,,l:] ;x: t ;] t-:]L+z - 19] 15 [ -30 -]zf,,,t ;] ,o,[ :,],": [-:] ,t :]L 3,. "|1': :: ;] ,|_1] ;-2,[-;] [ ;] -:|1] |_;] [;] 19, c :lr: ::] I'y,,: - I5y,2: 5, x : [ll ll] y, hyperbola 21,c: [-; -1] 5yí-5y,,:0,x: l:# :,Í]y,straightlines 23' C : l; :], yt2 * 5yz2 :10, x : |_;; *::l r, "rrin," 25, c: [-: -:], Jy - 5y,, :35, x : [_'r:,3'r:#]y, hyperbola 27,c:|'r: 'r:] 28y" - 4yz2:112,-: ['r:# 'r:,Yr7y,hyperbola A23 A24 App. 2 Answers to Odd-Numbered Problems Problem Set 8.5, page 36t s. 1anclT : e'E'Á- : C-1(-B)A 5. Hermitian,3 + :/1,|-t l - \,5]';3 - xE, L-t 1 + 7. Hermitian, unitary, 1, [1 i - i\/rl-; -1, [1 i + i\/žlT 9. Skew-Hermitian, 5i, tl 0 0]-, [0 1 1]-; -5i, [0 1 11. Skew-Hermitian, unitary, i, [1 0 1]-, [0 1 0]'; -i,11 13. Skew-Hermitian, -66i 15. Hermitian, 10 Chapter 8 Review Questions and Problems, page 362 ,|-,o -]] ^ [f -ttz 11. l L 2l3 fz 1 19 9 I 13.1t 0 I [o -3 |- - 1.0 15. l L 4.8 19. I.Iytz -| y22 : 1, ellipse Problem Set 9.1, page 370 1.2, -4,0; \,6; ttl\/Š,-ztl/Š,ol 5. -8, -6, 0; t0; [-0.8, -0.6, 0] 9. (+. 3.Zl, Ý118 13. |4, -2, O], I-2, I,0], [- I, *, 0) 17. [28, -14, -14] 23. (5.5.5.5, 0), (*,+, Ť) 27. |-8, -2, a]: r/u 31. [-9, 0, 0], I0, -2,0], [0, 0, -11]. Yes. |-25 25-l_t_20 2o1:35.| -, -L\/2 \/2) L \,5'\/ž) Problem Set 9.2, page 376 3. - 1, 0, 5; xDa; = Il\/%, 0, 5l\D6l 7.(7.5.0): V'm 11. (0. l. ll: r,51Ď 15. [10, -5, - 15] 19. I-2, 1, B], 16, -3, -24] 25. |0,0,9];9 29. v : [0, 0, -9] 33. lp*q-tul <6.Nothing 1-45 5l l _,_l sl.|rVtzsina) L \/2 v2 _) 7.I1 15. Use (1) and |cos 7| < t. i (a. a - Za.b + b.b) : Zlal' + 2|b|2 23. Orthogonality. Yes Ý2l' - 1]' 0 -1]' ,o]]:[;;] []i]:[:;] t o, ,, ,^1: t: : L-, 4 -,] L. 0 2l3f ln -113) 0l ;lo -+] 0l | .r.-, 5.0_] ;] ; ^;^,:,,] ,,L: ,4,0, -2 1.4 3. \/rn 5. |I2, -8, 4], [- 18, -9, -36] 9. -4,4 11. -24 17.|a + bl'+ la - bl': a.a .l_ 2a.b + b.b 19.0 2l. 15 App. 2 Answers to Odd-Numbered Problems 25.2,2,0, -2 27.19.1l. 31. 54.14' 33.54.79",7g.IIo,46.10" 37.3 39. I.4 41.If lu| : lnl or if a and b are orthogonal Problem Set 9.3, page 383 1. [0, 0, - 10], [0, 0, 10] 7. -20, -20 13. [10, -5, - 1] 19. -20, -20 27. [I, -I, 2] x [1, 2, 3l : 29. [0, 10, 0] x l4,3, 01 : 31.|t7,0, 0] x [1, 1, 0]l : 35. [18, 14,26];9x -| 7y * 37.16 7 . t\/1 137: c,9.4 + ].8 + 13.0 - 92: c 39. c : 2.5 A25 3. [-4, -8,26] 9.240 15.2 25. |-2,2,0l x |4, 4, Ol : |-7, - l, 3], \/59 [0, 0, -40], speed 40 29.82.45. 35. 63.43", I16.5J. 5. [0, 0, -60] ll. L19, -27,24], \/hn 17.30, -30 - 16k, 16 Problem Set 9.4, page 389 1. Hyperbolas 3. Hyperbolas 5. Circles 7. Ellipses;288, 100,409 elliptic ring between the ellipses x2 y'|--r (213)2' (ll2)2 -t 9. Ellipsoids 23. |8x,0, yzf, |0,0, x7], and 11. Cones 13. Planes [0, 18z, xyl l0, z, !], Lz,0, x], a, x,0] ,Y' "ŇlY':L Problem Set 9.5, page 398 1.|4+3cos/, 6+3sinr] 5. [3, -2 + 3 cos /, 3 sin r] 9.1\/Ž cos /, sin /, sin r] 13. Circle (x - 2)2 + (y + 2)2 : 7, z 3.|2-t,0, 4+tl 7.|a-|3t, b-2t, c+5tl 11. Helix on (x - 2)'+ (y - 6)2 : ,' -5 15.xalyn:1 17. Hyperbola rry : 1 23,r':[-5sin/, 5cos/, 0],u:[-sin/, cos/, 0],q: |4- 3w, 3*4w,0] 25. r' : [sinh /, cosh r], u : (cosh 24-t, [sinh r, cosh r], q : t3 l 4w, ! + swl 27. \Ei : cosh t, ( : sinh 1 : I.I75 29. Start from r(t) : |t, í(t)]. 33.v:r':[1, 2t, 0], lvl :\E+4F,fl:[0, 2,0] 35. v(0) :2alRi, a(0) : -rDzRj 37. I year : 365.86400 sec, R : 30 .365.86400/2rr : : |v|ztR: 5.9B . 10-6 [km/sec2] 151 , 106 km]. |u| : ,'R 39.R:3960 + 80mi:2.133. 107ft,s: lal : o2R:|v|zlR,|v| : vln: V6ó1 . 1ď : 25]00 [ftisec] : 17500 tmph] 43.r(t): |t, y(t), 0], r' : [l, !', 0], rr.r' : I + y'2,rrr: [0, !'', 0], etc. 47.3l(I + 9 + 9ta) A26 App, 2 Answers to Odd-Numbered Problems Problem Set 9.6, page 403 1,. w' : 2\/ŽGíth 4t)l(cosh 4t)ll2 3. w' : (cosh /)sintr -111cosh2 r) ln (cosh r) * sinh2 r) 5. w' : 3(2t4 + t&l'218t3 + 8r7; 7. e4u stnz Zu,leau sin 4u 9. -2(u2 + u2)-3u, -2(u' + u2)-3u Problem Set 9.7, page 409 1. |2x, zyl 3. flly, -xlyzl 7. L6, 4, 4] 9. |-1.25, 0] 13.I-4, 2] 15. [- 18, 24] 19. |6, 4] 21,. |-6, -I2] 23. [-0.0015, 0, -0.0020] 29. í8,6, 0] 31. [108, 108, 108] 35. 7l3 37. 2e2 tl/ t3 39.}x2 + Ey' - 2z2 41. xa + y3 - 3z2 Problem Set 9.8, page 413 I.3(x + y)' 3.2(x + xz * z) 7. 9x2y2z2 Chapter 9 Review Questions and 11. [- 1, 9, 24] 15. [0, 0, -]40], |0, 0, -740] 19. -495, -495 23.If u X v : 0. Always 27.3.4 5.|y -l 2, x - 2) 11. [0, - el t7. |48, -36] 27. fa, b, c] .\/ň Problems, page 416 13. 0, |-43, 54, 3l, |43, -54, -3] 17. |-24, 3, -398], íIl4, 95, -16] 21.90o,95.4" 25. |uy uz, -3l 29.If y > Lr,L, 5.(y+x+l)cosry 9.lut,Uz,Usf : r' : lx',y', zll: b,0, O], z' : O, Z: C3,y' :0,! : C2, x' : ! : c2, x: c2t l c,. Hence as / increases from 0 to 1, this "shear flow" transforms the cube into a parallelepiped of volume 1. 11. div (w x r) : 0 because uy l)2,u3 do not depend on x, y, Z, respectively. 13. (b) (fu), + (fu)u + (fu)" : fl(u1.)* -| (uz)u + (ua),] l f*ut t íuuz* f,ug, etc. (c) Use (b) with y : V8. 1,5.4(x + y)l(y - ň3 I7.0 19. e"a"(J'z' + ,'Z' l *'y') Problem Set 9.9, page 416 1. [0, 0, 4x - I] 3. [0, 0, 2e" sin yl 5. [0, 0, -4yl(x2 + y1l 9. curl , :7-2z, 0, 0], incompressible, v : r' : Ix', !', Z'f : [0, z', 0], x : C1, Z : C3,y' : z2 : C32,y : cg2t -l c2 11.curl y: [0, 0, -2],íncompressibla,x| :!,!' : -x,Z' :0,Z: cg, ydy+xdx:O,x2+y2:c 13. Irrotational, div v : 1, compressible, r: fclet, c2 -t, cset) 17.0r0rlxy - zx, yz - xy, zx - yz] 19. 0, 0, O, -2yz' - zzx' - Zry' App. 2 Answers to Odd-Numbered Problems 33.4516 37. 0, 2y' + (z -| x)2 41.0,2x2 + 4y'+ 2z2 + 4xz Problem Set l0.2, page 432 1. sin xy, 1 3. -$e-<"z+a2), O 7. ,'y f cosh z,392 11. sinh ac 15. ce" - aeb 17.1a2bc2 Problem Set l0.4, page 444 1,2x3y-2ry',81 - 36:45 5. e*-a _ e**a, _*r' + i"' + ,-' - á 9. 0 (why?) 13. y from 0 to lx, x from 0 to 2. Áns. cosh 15. y from 1 to 5 - x2. Ans. 56 35. No 39. [- 1, 1, - 1l, |-2z, -2J, _2y] 43.488/\E3n 45. 0 A27 Problem Set l0.1, page 425 1. F(r(r)) : 7125t6, t3, 0], 16448/7 : 2350 3. 0 + 160 5. r(r(r)) : [cosh t sinh2 t, cosh2 r sinh t),93.09 1.F(r(r)) : |t, cos /, sin r], 6n. 9. F(r(r)) : [cosh }r, sinh }r, etlg1,0.6857 11. F(r(r)) : ler, '', ,r'f, ,' + 2e4 - 3 15. 1113 17. [36n, á(s".)r, 36n] Problem Set l0.3, page 438 rI 3. l lx - x3 - (*' - í5)]dx : b, 5.} cosh 6 - cosh 3 + +Jo - )-á a 14 .d2 7. l *(e3" - n-")dx-áu" +žr-n- ? g. l (r.''ucos}-cos y)dyJo '' o- 2 ,o f1 ,r. l, (2x2 + 3) a, :8 ,.. ťf ^rn= d.y dx : ! 15. ' :3b,y : *h t7. It: bh3ll2, Iu : b3h/4 19. I*: (a * b)h3/24, Iu : h(aa - brl48(a - b)) 5. e*" * y, -2 13. No 19. No 3.3x2 * 3y', I875rrl2 : 2945 7.2x - 2y, -56115 11. Integrand 4. Ans. 40n 2-+sinh2 19.4e4 - 4 :e-2 Problem Set l0.5, page 448 1. Straight lines, k 3.x2/a2 -l y2lP2:l,ellipses, straightlines, |- bcosu, asinu, 0] 5. z: (cla)Ýx2 ,| y', circles, straight lines, f-acucos u, - acusin u, azu] 7. x2l9 + y2ll6: z, ellipses, parabolas, [- 8u2 cos tl, -6u2 sin u, IZu] 9, x2/4 + y2/9 + z2/76: 1, ellipses, [12 cos2 u cos Lt, 8 cos2 tl sin u, 6 cos u sin u] 13. |I\u, 10u, 1.6 - 4u -| 2u], |4O, -20, 100] 15. |-2 -I- cos l) cos l,t, cos u sin z, 2 * sin u], [cos2 I) cos tt, cos2 u sin u, cos u sin u] A28 App. 2 Answers to Odd-Numbered Problems 17. |u, l), 3u21,10, -6u, 1] 19. |u cos u, 3u sín u, 3u], [-9u cos u, -3u stn u, 3u] 2l. Becalse r?/ and r, are tangent to the coordinate curves U : const and u : const, respectively. 23. |i, ú,ň2 + D'], Ň : |-2i, -2ú,1] Problem Set 10.6, page 456 3. -18 9.*o" 15. 140\/613 17.I28Ťr\/ž/3 : 189.6 t9. L*rr2(37zlz - 5312.1 : 22.00 27. thah/Ž 29. rrh + 2nh3l3 Problem Set l0.7, page 463 I. -64 1 a-l. L,ll 'l,.8a3b3c312] 7. 234rr |3. rrhíll\ 21.0 11. Exact, -542l3 17. By Stokes, + 18zr 23.0,4al3rr 29. By Gauss, 100zr 35. Direct, 5(e2 - I1 3,6 9.2a5/3 17. I08rr 23. 8 5. -I28n 11. I7hl4 25.2rrh 5. azlrr lt. haarrl2 19.216rr 25.384n 15. By Green, II52rr 21.4l5,8l15 27. Direct, 5 33. Dírect, nh Problem Set l0.8, page 468 1. Integrals4, 1, I (x: 1), 4,I,1 (y : 1), -8, 1, l (z : 1), 0 (x : y : z : 0) 3. 2 (volume integral of 6y2), 2 (surface integral over.r : 1). Others 0 5. Volume integral of 6y2 - 6x2 is 0. 2 (x : 1), -2 (y : 1), others 0. 7,E : |x, y, z], div F : 3, In(2), Sec. 10.7, F.n : |F||n| cos @ : \/ť + y\ ,' cos @ : r cos @. 9. F : [x, 0, 0], div F : 1, use (2*), Sec. 10.7, etc. Problem Set 10.9, page 473 1. [0, 8z, 16].[0, -I, If, -+-I2 3.I-e", - *, eal.f-l, -I, Il, +@2 - I) 5., : [z, l), u2], (curl F),N : -4ue2", t(4 - 4r') 7. (curl F).n :3l2, -r3a2l2 9. The sides contribute a,3a2l2, -a,0. 11. curl F : [0, 0, 6),2411 13. (curl F).n - 2x - 2y, Il3 15. -rrl4 17. (curl F).N : zr(cos rrx l sin rry),2 19. F.1 : [-sin 0, cos 0].1-sin 0, cos 0] :1,2r,0 Chapter 10 Review Questions and Problems, pa1e 473 13. Not exact, ea - 7 19. By Stokes, +12r 25.8l7, II8l49 31. By Gauss, 40abc App. 2 Answers to Odd-Numbered Problems 15. Translate by l. 17.Setx:0. Problem Set l1.1, page 485 3.2rrln,2rrln, k, k, kln, kln 13.1 *Z(, l l \ 'u, ' -; \cosr-,cos3l * šcos5x - - ) É.+*!( l l ...\"'r-; \coSJ*, cos3x* xcos5x+, ) 17.!_Z( l t \ '',T-; \cosr* rcos3x+icos5x* ) t sin - - : sin2x- + sin 3x - + . . . 4l l l \ 19.-a(."r** g-cos3x* xcossx- ) *r(,'n"* jsin3x+sin5x- ) ,r. + rr2 -4(.", - - + cos 2x; cos 3x - - ) n.!nr_! 1 4 1 1J. 6,, n cos-rrcos2x* ,^ cos3x* , cos4x 29,f' :2x,í":2, jt:O,j!: -4r,j'|:0,an: * ( I)r-^"cosnrr,etc. Problem Set 11.2, page 49O 1.4 Í.in3{ , l :__3rrx l 5rrx .. \l' Ťr \""' 2 Ť;sln 2 * -sin , *, ) .. + - +(.o,,,," - + cos2rrx; cos 3zrx _ - ) S.Rectifier.Z 4 / l ^ l ' ,6r,..+...\Tr-; (,* cos2t,u* r., cos4rrx+fr,o t 7.Rectifie..+- 1(r"rrrx1l t \.. r\vlllllvlt 2 rr2 \vvJ,J^ . 9 cOs 3rrx l ; cos 5rrx + . . .) ' : - +(.or,," - + cos2ltx - + cos 3rx* cos4rrx- - ) ", 1 + (*, Z - + cos .Tx+,o, ! - * "o, !- * cos3rrx \ -) 31 1 'a. * *, cos2x* s cos4x A29 App. 2 Answers to Odd-Numbered Problems Problem Set lI.4, page 499 3. Use (5). oo ( - 1\n - nlL:-@ n+o Ť1 n!L:-@ n+o n -2i Š l ^(2n-Ilitl.-- .aTr zn+ I ?L:-@ *2 o" (_ l." 3-n" 7L: *@ n+o A30 Problem Set l1.3, page 496 1. Even, odd, neither, even, neither, odd 3. odd 5. Neither 7. odd 9. odd n.!+4 l/.or"+ lcos3x+ l cos5x* \ 2 Ť \ 9 25 l 4l t l \ 13. - lsinx sin3x + sin5x - +,,,| Ťr \ 9 25 l 4/ tx l 3nx l 5rrx \ 15.1 _ _ lsin _sin_ + _sin_ +...I Ťr \ 2 3 2 5 2 l 4l nx I 3rrx I 5nx \ 17.(a1 l,(b)-|sin ^ -r=sin ^ *=sin ^ +",I Ťr \ 2 3 2 5 2 l 8l Ťrx I 3rrx I 5nx \ 19.(a)r* ď |,"orr*rcos , * xcos , +...) 4/ rrx 1 1 3rrx 1 \ (b)-|sin ^ -l ^sinrrx*;sin ^ +-sin2rrx+",| Ťr \ 2 2 3 2 4 l 3 2 / ,* l 3rrx l 5rrx l 7rrx \ 21.h\ -lcos -cos-+-cos -cos-+-...l2 7T \ 2 3 2 5 2 7 2 l 6l ,* 1 1 3rrx I 5nx 1 \ (b)-|sin ^ -=sinrrx*;sin ^ i;sin _ -Tsin3nz +",lŤr \ 2 3 3 2 5 2 9 l L 4Ll nx l 3rrx 1 5nx \ 23.h\ _ lcos- + -cos- + -cos+...l2 Ťr" \ t 9 L 25 L l 2L / rrx I 2nx l 3rrx \ (b) _ lsin_ sin_ + _sin_ _ + ...l Ťr \ t 2 L 3 L l Ťr 4/ t 1 \ 25.(a); + ] |.o, x+: cos3x + + cos5x + ",I2 Tr \ 9 25 l ltl\ @2lsinx -| = sin 2x -| = sin 3x +,,,| \23l App. 2 Answers to Odd-Numbered Problems Problem Set lt.5, page 5Ol 3. (0.05n)2 in Dnchanges to (0.02n)2, which gives Cs:0.5100, leaving the other coefficients almost unaffected. 5.y: c1 cos olt l c2 sin at l A(a) cos /, A(a): ll(az - 1) < }if o < 1(phase shift!) and ) 0 ífa2 > t 7, y : c1 cos at * c2 sin 0)t + Žr#+ cosnt 9.y: c1 cos at * c2 sin at * ]= * ! (_] Il9 \ Lz Jtll wt 2r' ; \;= cos / i d,= cos 3r +...| / 1l. l, : c1 cos at l c2 sin at l * - "TF= cos 2r l -:--:.- ,--- _ cos 4r 5,5(a' - 16) 13. The situation is the same as in Fig. 53 in Sec. 2.8. 3c8I . _\' : - -;-. ---;-; cos 3r - --,----- - sin 3r' 64 l 9c' 64 -l 9c2 orrl Jl lí/ l7.r,: ž (_ ncb" (l - n2)bn . \ n_7 \ D" cos '?r * a sin nt),D,: (| - n2)2 + n2c2 19. I(t): Ž (An cos nt l Bn sin nt), An : (_l1n*t a#oĎ , Bn: (-1)n*1 #, Dn: (10 - ,r)' -ťl00n2 Problem Set lt.6, page 5O5 / l.F:2(rin*_ | (-l)^u' ,._ \ I., -\o,,, sin2x + ... * -Ť- sin Nx/ , E* :8.1.5.0, 3.6,2.8,2.3 Tr 4l l 1 \ 3,F:, - 7 (.or"*;cos3-r +;cos5x - ).E*:0.0748,0.0748. 0.01 l9, 0.01 19, 0.0037 2 4/l l l .\5'É-: ; - ; l, " cos2x* :., cos4-t + ;cos6xf " l,E* : 0.595I, 0.0292, 0.0292, 0.0066, 0.0066 4l l l \ 7.F:7 (ri"** a sin3x+ , sin5x - ),E*:1.Ig02, 1.7902,0.6243, 0.6243,0.4206 (0.1272 when N : 20) ^8/ l l \ 9. - lsin -r,+ _ sin 3x + ;; sin 5x + . . .l.e* :0,0295.0.0295.0.0015, Ťr \ 27 -'^' ""' l25 Jlr! J-. 0.0015, 0.00023 l I I i..,, A32 App. 2 Answers to Odd-Numbered Problems 5. Use f : (rrl2) cos u and (1l) in App. 3.1 to getÁ : (cos (rrwl2))l(I Problem Set l1.7, page 512 ^oo t. í(x): Ťre-* (x > 0) gives Á : l ,-" cos }4/u du : (see Example 3), etc. Jo 3. f(x) : tre-" gives A : Il(I + ,'). 2 r" sin aw cos xw i.- l -r]r, Ťť"o W 2 r-cosrrw*l 11.- l ^ cos.rtt,dtl rrJo |-w. 2 r- cos w l w sin w -lo l lr Jn ^,2 ,B: I-|w2 - w\. 1 cos xw dw 5. \/ nD ,-" (x > 0) 9. Yes, no 1 Ilw" 2 r- 7Tw - sin rrul l7.- l rr Jo w2 sín xw dw 2r- 15.- | ,,ll "O 2r- 19.- l rr Jo nJo w2 stn rrw j sin xw tlw I-w' wa - sin wa sin xw dw w' Problem Set l1.8, pa8e 517 _ lí |,i"2y ]:-,\1. l- | ''V,\ , l 7. \/ŤrDcos l,ť if 0 < w { rrlLand 0 if w > nl2 Ea217. J"(í' ) : .!"l-uít : -utr"(f) - - i - = " /, V Tr a2+,2 l/. w .T,"(ft: i- _'s\J' V Tr a2 + w2 19. In (5) for í(ax) set ax : u. t|. \/'|2la wllw2 + n2) 13.'5r(xe-r2l21 - 9r(- (e-*'tzr') : ,'#"(e-"'tz' : we Problem Set l1.9, pate 528 3. ik(e-ib* - Dl(\/-2Ťw) 7.|Q + iw)e-i* - |l(Ýňw21 : w?F,(í) - 5. \relŤr)k (sin w)lw 9.\/(Urr)i(cos w - I)lw r; l1,t. \Ťr 11.\e--'n 13. 1eib- - e-ib1,)/(twt/-2rr) : \/-2/r(sin bw)lw Chapter ll Review Questions and Problems, page 532 4k/ t 1 \ 11. lsin rrx + - sin 3rrx + - sin5rrx +,,, l Ťr \ 3 5 l Ťrx 2- l3x1l5x -sln -sin4.r* -sin-32452 3:r.x l 5rrx \_l_sin r...l 2252l l sinxi 2 /,13. 4(sin ; rs. 1(.i, 24 ",;-; --) 1 - Sln 9 lt| cos l6rrx l \1,3 11 cos 32rrx + cos 48rrx i3.5 5.1 ) App. 2 Answers to Odd-Numbered Problems rr2 111 'n. i - cos 2x -Ť 7 cos 4x -n cos 6x * f .o. 8_r - + . . . 2l. rrl4 by Prob. 11 23. rr2l8 by Prob. 15 l1 'r., [/(x)+ f(-x)l, 1Ií(x) -/(-x)] 8 27, rr ír 29. B.l05,4. A33 l*l3xl5x\ \'o'r* g cos7* x cos7+ ,) 963, 3.567,2.781,2.279, I.929, L6]3, I.4]1 3l.y : cr cos .,,,/ t czsin ol - + - 4( ;:" I cos4r \ "' \t,-''- | - 16 ' dr- |6 T - "') cos 2t ' d'- 4 cos 3r1 +-9 c -g 1r*33.- | rr Jo 2r- 35.- | ,.ll "O 4r- 37.- l Ťr Jo problem set I.kcos2ml Sft/ 3. - (cos 4/ 5' *, ('o, )/_ 7. z (rvz I- + 9íV2 (cosly * r.l,sinw - 1)coswx * (stnw - } cos w)stnwx W2 w-sinwcosw sín wx dw sin 2w - 2w cos 2w cos wx dw dw W2 w3 w2+4 Problem Set l2.1, page 537 l. u : cl(x) cos 4y + c2@) sin 4y 5. u : c(x)g-u -| e"al(x -| I) 9.u:cl@)yic2(x)y-2 15. c : I/4 19. rrl4 27. u: 110 - (110/1n 100) ln (*' + yr) 3. u : c{x) -| cz@)y 7. u : c(x) exp (}y2 cosh x) 11.u: c(x)ea + h(y) 17. Any c 2l. Any c and al 29.u:clx+cr(y) l2.3, page 546 sin 2nx .1 1\rrt sin rrx l Z7 cos3rrt sin3nx * * cos 5rrt sin slrx + . . .) .1 1\1Tt sln nrx - = cos 3rrt sín 3rxg cos3rrt sin 3zrx + ; cos 5rt sin 5rrx - - ) - 1) cos Ťt sin zrx - + cos 2rrt sin Zrx i l) cos 3zrr sin :rr-" - . . .) l 39. rYŤr A34 )l _ 9. + lrZ - Ý21 cos rrísin Ťrx -Ťr' \ I- Q + X/2\ cos 5zZ sin 5rrx -|- 25 7- 9 Q + \/2) cos 3rt sin 3zrx * ) 17.u:#(.",[.(;)',] ,ini-+.o,[. (+)',] ,''' +- ) 19. (a) u(O,t):0, (b) u(L,t):0, (c) u*(0,t):0, (d) u*(L, /):0. C: -A, D : -B from (a), (b). Insert this. The coefficient determinant resulting from (c), (d) must be zero to have a nontrivial solution. This gtves (22). Problem Set l2.4, page 552 3. c2 :300/t0.9/(2,9.80)] : 80.832 7m2lsecz1 11. Hyperboltc, u : ft@) + f2@ + y) 13. Elliptic, u : íl(y + 3ix) + f2Q - 3ix) 15. Parabolic, u : xf t(x - y) + íz@ - y) 17. Parabolic, u : xf {2x + y) + f2(2x + y) 19. Hyperbolíc, u : (Ily)f{xy) + fz!) Problem Set 12.5, page 560 5. u : stn 0.4nx e-L,752{6Ť2tLoo 7. u: Z (Zsin 0. |.Tx e-.o,o1,752n2t * + sin0.2tx e_ool752e,," _ . . .) Ť \rr" 2 l 9.u:+ (r,"o. lnxe--o.ol7u,2l2t-; sin0.3zrx e--o.o7752*lfl2t - ) t'l,. u : L\ t u11, where LIII : u - ut satisfies the boundary conditions of the text, so Ť n ŤrT ) rL nTrX thatu11: ' B,, sin "T n-'"nt/Llzt. B,: ; I, Lfa) - u16)],in ff a, 13.F: l"or"]}* B sin px,F'(0): Bp: O,B: O,F'(L) - -Apsinpl:0, p : nnlL, etc. 15. u: I I7. u: T + +(.o, x e-' - +cos 2x e-4' + } .o. 3x e-.g' - - ) rr2 l _^ l t9.u:; * cos 2xe-at * i cos4x e-l6t * g cos6x e-36t + ", 23. - + Ž nBne-^n't 25.w: g-Ft L n.:7 27.ur- c2u**:O,w": -Ne-o*lc2,w: ,*L [-.-"" - +(l - e--tr* r] so that w(0) : w(L) : 0. 29. u: (sin }rrx sinh lrry)lsinh n 31,. u: 'O ;Ť n:1, (2n - 1) sinh (2n - I)n . (2n - |)rx (2n - l)rry sin ^ slnh u- App. 2 Answers to Odd-Numbered Problems @ 33.u:Aox+2 e* n:1 A35 sinh (nrrxl24) sinh nrr nŤry cos 24 ' .lr'nlr24nnAo: ,F J" fll',) dy. An: " J, /(y) cos } a 35.i AnsinTsinhnrr(b-D ,An: ,,' n:I a a " asinh(nrrbla) [".f(") ,rn 'o nŤrx a Problem Set 12.6, page 568 2 sínap 2 so sin ap I. A : *r, B : 0, u : ; J, Ť cos pJ e-"'P'' dp 3. A : e-P, B : O, u : [-ao, px e-P-czP't d.p 'o 5.Set Ťrl):s.A: lif 0 1plrr 0 21.6z2k3 + i) I lo 35. |z1 4 zr|' : (zt * z)(íl+ zr) : (zt + z)(4 + Re ztZz = lzrZrl (Prob. 32): zlž1l zlZ2 l ^7r7t 1- zzZz : lzll2 -| 2 Re zlž2l lrrl' = lrrl' + z|zt||zrl + lzr|' : (lzrl + lzrD', Take the square root to get (6). Problem Set l3.3, page 6'17 1. Circle of radius $, center 3 + 2i 3. Set obtained from an open disk of radius 1 by omitting its center z : I 7. y-axis A37 ,) 1. Yes 7. No 13. z212 19. No 27.Use (4), (5), and (1). 3. No 9.Yesforz*0 15. ln |zI + i Arg z 21. No 5. Yes 17. z3 23. c : 1, cos x sinh y ll. r*: xlr: cos 0, Tu: sin 0, 0* : -(sin 0)lr, 0r: (cos 0)/r, (a) 0 : u* - uu : llrcos 0 * ur(-sin 0)lr - u, sin 0 - uu(cos 0)/r. (b) 0 : uu l ur: ursin 0 1 ur(cos 0)lr -| u, cos 0 1- ur(-sin 0)lr. Multiply (a) by cos 0, (b) by sin 0, and add. Etc. Problem Set l3.5, page 626 3. -1.I3I20 + 2.47I73i, : 2.71828 5. -i, 1 7. eo,B(cos 5 - i sin 5), 2.22554 9. e-2* cos2y, -e-2* sin2y 11. exp (*' - y') cos 2xy, exp (*' - y21 sín 2xy 13. einla, e5Tžl4 15.'(,G exp [i (0 -l 2kn)/n], k : 0, . . ., n - 7 17.9eň 19. 7 : ln 2 * ri -| 2nri (n : 0, +l, 2l. z: ln 5 - arctanf;t * Znrt (n : 0,1, . . .) Problem Set 13.6, page 629 3. Use (11), then (5) for eia,and simplify. 5. Use (11) and simplify. 7. cos 1 cosh I - i sin 1 sinh l : 0.83373 - 0.98890i A38 App. 2 Answers to Odd-Numbered Problems 25. Insert the definitions on the left, multiply out, simplify. Problem Set l3.7, page 633 1. ln1O*ni 3.}h8-1rri 5. ln 5 i (arctan t - r)i : 1.609 - 2.2I4i 9. ]4.203,74.210 13. -1 17.7: *(2n+ l)Ťril2 2'l,. z: +n7Ti 7.0.9273i '1.+-(2n+ I)Ťri, ft:0, l,... 15. (rr - 1 r- Znrr)i, tr : 0, 1,,,, 17. |n (i') : (*2n + I)Ťri, 2In i : 19. eo,s(cos 0.7 -| i sin 0.7) : I.032 2I. e21I + i)l\/Ž 25.2.8079 + 1.3I79i 1l. -3.7245 - 0.5I1,82i 15. cosh 4:27.308 19. z : l1zn + I)n - (- 1)'1 .4436i 9.}h2-|ni 13. ln6l-(2n-ll)ni, fl:0, 1,... *(4n+I)Ťri, ft:0, 1,", + 0.870i 23. 64(cos (ln 4) * l sin Qn a)) 27. (I + i)l\/' Chapter 13 Review Questions and Problems, page 634 Problem Set 14.1, page 645 1,. Straight segment from I + 3i to 4 1, 1,2i 3. Circle of radius 3, center 4 + i 5. Semicircle, radius 1, center 0 7. Ellipse, half-axes 6 and 5 9. Parabolay : }x3 from -1 - tt to Z + +t lI.e-it(O 2 1,1.2rr2i 13. -2nirlrl2 rrie"l2lz4 if |a - 2 - 5.8Ťi 11. rri 5. tia3/3 9. rt(cos } - sin }; il<3,0if|a-2-il> 21. - rri 27. rr 29.0 Chapter 14 Review Questions and Problems, page 662 17. -6rri 23.li sin 8 19. 0 25.0 Problem Set l5.1, page 672 1. Bounded, divergent, + 1 5. Unbounded 7. Bounded, divergent, -| tlÝŽ -+ i,0, I, 17. Convergent 21. Conditionally convergent 3. Bounded, convergent, 0 -2 19. Divergent 23. Divergent by Theorem 3 9. Convergent, 0 I3.|z- - /| < *r,|rŤ. - /*l N(e)), hence lrn+ zŤ.- (l + l*)l < le + le 27. n:_J_l00 !_]sil : I25 (why?); Itoo + 75i|l25lI25! : I25l'ul|\/25ť, Q25le)r25] : gtzs1\/N}Ťr : 6.9I .lO52 Problem Set l5.2, page 677 1.2 anz2n :2 onk\n,lz'| { R : Iim|anlan*r|, hence lrl . Ř. 5. -1, eby (6) and (1 * Iln)" ---> e.3. -i, I 7.0, |bla| 13.3 -2i, I 9.0, 1 4- 15. i, 1l\/2 11.0, 1 17. 0, \/' Problem Set l5.3, page 682 l. 3 3. \/' 7.Il\/j g. 1 Problem Set l5.4, page 690 1. 1 - 2z,| 2z2 - tr" + |rn - + 3. e-2ž1I + (. + 2i) + ž(,+ 2i)2 + 5. 1-1rr-Žň'+hrr_*Ďnl.L + lr + lilz - ,) + Gi + žilrz 5. \/5/3 "', R: oo á(. + 2t)3 + kk + 2i)4 + . . ., R : oo #rr-*ňu+-", R:oo - i)' - irr- i)3 + - ., R : \/ž A40 App. 2 Answers to Odd-Numbered Problems 9. l - žr'+ *ro - *'.u + #,.u - +,,,. R : :n 1I.4(z- 1) + 10(. - D2 + 16(, - 1)' + I4(, - 1)n + 6(z - 1)' + (z - 1;6 B. Ql{flk - z3l3 + z5(2|5) - z't13|7) +. . .), R : oo 15.73l(1!3) - z'l(3tl) + z|Il(5l11) - + ..., R: * 19.z-|*.'+#ru+#r' + ,, R:žn Problem Set l5.5, page 697 1. Use Theorem 1. 5.1z"| < 1 and 2lln2 converges. 9.|z+I-2i|šrlR:4 13. Nowhere 3. R : Il{rr > 0.56 7.|tanbn 1.1l = l, Il(n2 -l I) < tln2 11.|z| <2- s(6>0) 15.|z| <\E - 6(6>0) 13. 1, } I-nllt + z)l(I - z)] 17. l/{rr, lI - n(z - 21zYt Chapter 15 Review Questions and Problems, page 698 1I. *, e3" 15. oo í9.I/3 21.-I - (r- Ťri)- (z- ri12l2! R: oo 23.*+ž(, + 1) +*(. + 1)2 +#(.+ 1)3 +..., R:2 25. I +3z+ 6z2 + 10z3 + .,R: l.Differentiatethegeometricseries. 27.i + (z+ t) - i(z+ D2 - k+i)3 + ..., R : 1 29. -k - !rr) - + k - !r)3 - + k - !rr15 + - . . ., R : oo Problem Set 16.1, page 7O7 ll L " i . Z,= Z." 11 3..-ó7u 7o!d 11 5. ^* Z," Z.' > _,| e. -e l. -11,- L 9. 11. ž( R:2 co - žt.n-O : l ;L + a J 13.- _ +2+(az- I 1) App. 2 Answers to Odd-Numbered Problems A41 1.1 ,t , 1.1 >1 21. (1 co B.> n-O 4 al, 1l - ^džz4n, |z| < l, (-l)n*'(z -f lrr;zn-t ž)ž l ,* ,1.1 > 1 1. i,4i 5. 1/5! (at 7 : 0; 9. -+(at 7 : 1), } (at z : _ I) 15. erl': 1 + I/z -| . ..,Ans.2ti 19. -4rrisinh }zr 23. 0 l7.27il3 23. rril4 27. 6ni 33. 0 žrl>o 19.5rr 25.0 (rz even), (-11tz 29.2rrl7 35. rrl2 3. -ii (at 7 : 2i), ii (at _2i) 7. l (at lnrr) 11. -I (at z : t*r, t|n, . . .) 17. Simple poles at *+,.Ans. -4i 2l. -4i sinh } 25. ž(at 7 : t\, Z 1at z : !). Ans. 5.2rrl3 11.2rrl3 17. nl2 23. rr 726 2l. }zr cos 10 2rril(n - 1)! (n odd) 31. 4rrl\/5 (2n)t Problem Set 16.2, page 711 í +1 _3t. !j, nž,. . . (poles of 2nd order), oo (essential singularity) 3. 0, +{rr, !ÝŤrr,, , , (simple poles), m (essential singularity) 5. oo (essential singularity) 7. -|I, +i (fourth-order poles), m (essential singularity) 9. -+i (essential singularities) 13. -16i (fourth order) 15. -| 1, *2,. . . (third order) 17" )-i/\/j (simple) 19. -|2i (simple), 0, -+2Tí, +-4ri,. . . (second order) 21.0, !2rr, t4rr,. . . (fourth order) 23. f(z) : (z - z n7(z), 8(z * 0, hence í'(z): (z - zg)2"sz(z). Problem Set 16.3, page 717 5Ti Problem Set 16.4, page 725 1.2rrl\/t3 3.2rrl35 7.0 9. ,7T 13. rrll6 15. 0 19. 0 21. O 25.0 27. -rrt2 Chapter ló Review Questions and Problems, pate t_ A42 App.2 Answers to Odd-Numbered Problems Problem Set l7.1, page 733 3. Only the size 5. X : C,W : -y -| ic,! : k,w : -k -| ix 7. -3rrl4 { Arg w 1 3rrl4, |w| < tt} 9. |w| > 3 11.|w| 0 15. ln2=u šln3, rrl4 O 11. Elliptic annulus bounded by u2/coshz I + u2lsinhz 1 : 1 and u2 lcoshz 5 + u2lsinh2 5 : I '3. -r(2n -l l)tl2, fl : 0, 1, . . . 15.0 { Im t 1 n is the image of R undeí t: z'. Arr. e' : e" 17,0, -+-i, -+2i, . . . 19. uzlcosh2 1 + uzlstnh2 1 š 1, u { 0 21.u<0 23. -l = u š l,u : 0 (c : 0), u2lcoshz c + uzlsínhz c : I (c + 0) 25,In2 < u šln 3, rrl4 < u < rrl2 Problem Set l7.5, page 747 1. w moves once around the unit circle. 5. -5l3,2 sheets 7. - ilz, 3 sheets 9. 0, 2 sheets App.2 Answers to Odd-Numbered Problems A43 Chapter 17 Review Questions and Problem s, page 747 1l.u:ir'- I,1u2 - | 13.|w|:20.25,|argw|< rrl2 15. The domain between u : i - t)2 and u : I - iu' 17.|w+*I:+ 19.u:I 2t.|argw| uk that has length k. 15. No; there is no way of traveling along (3,4) only once. 21. From m to 100m, IOm, 2.5m, m * 4.6 Problem Set 23.3, page 966 1. (1, 2), (2, 4), (4, 3); Lz : 6, Ls: 18, Lq : 74 3. (1, 2), (1, 4), (2,3); Lz : 2, Ls : 5, La : 5 5. (1, 4), (2,4), (3,4), (3, 5); Lz : 4, Lz : 3, L+: 2, L5 : 8 7. (|,5), (2,3), (2,6), (3,4), (3,5); Lz: 9, Ls: 7, L+: 8, |s : 4, Le : 74 Problem Set 23.4, page 969 1 s.+(-z L:70 3-5 2 1.|( ,3 L:12 4( 5 5.4 5. 1-2< .8 5 z(-e-+ ] L_ A54 App.2 Answers to Odd-Numbered Problems 2 g.1-3-4( L:38\ 5-6, 15. G is connected. If G were not a tree, it would have a cycle, but this cycle would provide two paths between any pair of its vertices, contradicting the uniqueness. 19. If we add an edge (u, u) to 7, then since Z is connected, there is a path u ---> u in T which, together with (u, u), forms a cycle. Problem Set 23.5, page 972 1. (1, 2), (1,4), (3,4), (4,5), L : 12 3. (1, 2), (2,8), (8, 7), (8, 6), (6, 5), (2, 4), (4,3), L : 40 5. (1, 4), (3,4), (2,4), (3,5), L : 20 7. (I, 2), O, 3), (1 , 4), (2, 6), (3, 5), L : 32 11. If G is a tree 13. A shortest spanning tree of the largest connected graph that contains vertex 1 Problem Set 23.6, page 978 1. 1 - 2 - 5, Lí: 2;I - 4 - 2 - 5, ^f : 2,etc. 3. 1 - 2 - 4 - 6, ^í : 2il - 2 - 3 - 5 - 6, ^f : I,etc. 5. f ,r:4, f13: l, fu:4, f +z: 4, f qs:0, fzs: 8, íss: I, í :9 7. f n: 4, f13:3, fz+:4, fss:3, f sq:2, í+a:6, fsa: I, f :7 9. {4,5,6},28 11. {2,4,6},50 13. 1 - 2 - 3 - ], ^f : 2;I - 4 - 5 - 6 - 7, L7 : 1, I - 2- 3 - 6 - 7, Lf : I; írn^r: 14 1,5. {3, 5,7},22 19. If f ,i < cii ds well as íe4 > 0 17. S: {1, 4},cap (S, D :6 + 8 : 14 Problem Set 23.7, pate 982 3. (2,3) and (5, 6) 5. 1 - 2- 5,Lt:2; 1 - 4 -2- 5,L,r: I; í: 6+2+ I :9 7.I -Z- 4- 6,L*:2; I - 3-5 - 6,A.: 1; f : 4+2+ I :J 9. By considering only edges with one labeled end and one unlabeled end 17. S: {1,2,4,5},T: {3,6}, cap (^, T): 1,4 Problem Set 23.8, page 986 1. No 3. No 5. Yes, S: {1,4,5,8} 7. Yes; a graph is not bipartite if it has a nonbipartite subgraph. 9. 1-2-3-5 11. (1, 5), (2,3) by inspection. The augmenting path I - 2 - 3 - 5 gives I, 2 - 3 - 5, that is, (1, 2), (3, 5). 13. (1 , 4), (2,3), (5, 7) by inspection. Or (1 ,2), (3, 4), (5,1) by the use of the path I-2-3-4. 15. 3 25. K3 23. No; K, is not planar. 11. Yes 19.3 A55 App.2 Answers to Odd-Numbered Problems Chapter 23 Review Questions and Problems, page 987 i l l "[;iil,.|| i I r0 l0l |o 0 l 17. I ll 0 0 I L000 2l. Vertex Incident Edges 23. 4 1 2 3 2,- 3 -I,3 'y-e2 25.4 29.1-4-3-2.L:16 ,,V 27. L2: 10, Ls: 75, L4: 13 33.7:7 Problem Set 24.1, page 996 1. qt: 79, Qru : 20, qu : 20.5 3. qr,: 38, Qtw: 44, q, : 54 5. qt : 69.7, Qlw : 70.5, qry : 71.2 ?.;; : 2.3, Qttl : 2.4, qu : 2.45 9. q" : 399, Qlw : 407, qry : 407 tl.I : 79.875,s : 0.835, IQR : 1.5 13,i:70.49,s: 1.047, IQR: 1.5 15.r:4O0.4,s:1.618, IQR:217.0 0 300 19.3.54. I.2g Problem Set 24.2, page 999 1. 4 outcomes: HH, HT, TH, TT (H : Head, T : Tail) 3.62 :36 outcomes (1, 1), (1 ,2), . . ., (6,6) 5. Infinitely many outcomes S, S",S, S"S"S, . . . (S : ..Six'') 7, The space of ordered triples of nonnegative numbers 9. The space of ordered pairs of numbers 11. Yes 13. E: {^, S",S, S"S", }, ff : {S"S"S"S, S"S"s"s"s, . . .} (. : ..Six,,) Problem Set 24.3, pase 1OO5 1. (a) 0.93 : 72.97o, (b) ffi.88.88 : 72.657o 3. #8 .á# . ffi.ffi .#E : 90.35vo 5. 1- *:0.gS 7. l - 0.752:0.4375<0.5 9. P(MMX[) + P(MMFII/[) + P(MFM^4) + P(FMMx/t): t + 3.# : * A56 App. 2 Answers to Odd-Numbered Problems 11. 3% + # - t : 38 by Theorem 3, or by counting outcomes 13. 0.08 + 0.04 - 0.08 , 0.04 : 11 .68%o 15. 0.954 : 8I.5%a t7.I - 0.914 -- II.SVo Problem Set 24.4, page 'l010 3. In 40 320 ways 7.2I0,70, Il2,28 11. (?r') : 635013 559600 15.676000 Problem Set 24.5, page 10l5 1. k -- 1/55 by (6) 5. No because of (6) ,2 9. P(X > 1200) : I ap.zs - (x -Jlz 1I. k : 2.5;507o 17. X > b, X > b, X 1 c, X š c, etc. Problem Set 24.6, page 1019 I.2/3, IlI8 5.4, 16/3 9. p: Il0 : 25; P -- 20.2%io 13. ]50, 1, 0.002 Problem $et 24.7, page 1025 l. 0.0625, 0.25, 0.9375, 0,9375 5.0.265 7. f(x) : 0.5*e-o,5tx!,1.101 + fQ) : e 9. 1-e-o'':I87o a- I2O 135 30 1 lJ, 'E6, 2S6,286, 286 Problem Set 24.8, page 1031 1. 0.1587, 0.6306, 0.5, 0.4950 5. 167o 9. About 23 13.t:1084hours Problem Set 24.9, pase 1040 1. 1/8, 3/16,3l8 5. (?ro) : 1t40 9.9|l(2!31.4l.) : 1260. Ans. Il1260 t3. 1184,5l2I 3. k : 1/8 by (10) 7.1- P(X= 3):0.5 1.5)2] dx - 0.896. Ans0.8963 : JZ?o 13. k: 1.1565; 26.97o 3. 3.5, 2.917 7. $643.50 tl.ž,*,(x - b\m 15. 15c - 500c3 -- 0.97, c : 0.0855 3.647o -o,s(1.0 + 0.5):0.91 .Ans.9vo 11. 0.991oo : 36.6Vo 3. I7.29, I0.7I, 19.I52 7.3I.I7o,95.57o 11. About 587o 3.2l9,2l9, Il2 5. ír(y) : Ll(Fz - az) if ar 1 ! < Fz and 0 elsewhere 7. 21 .45 mm, 0.38 mm 9.25.26 cm, 0.0078 cm App.2 Answers to Odd-Numbered Problems 13. Independent, ft@):0.Ie-o-1'if x> O, fz(y):0.Ie-o,lu if v > 0"36.8%a l5.507o 17. No Chapter 24 Review Questions and Problems, page 1041 2l. Qt : 22.3, Qru : 23.3, Qu : 23.5 23. r : 22.89,s : 1 .O28, s2 : 1.056 25. H, TH, TTH, etc. 27. f (0) : 0.80816, í(l): 0.18367, f(2): 0.00816 29, Always B E A U B.If also A C B,then B : Á U B, etc. 31.113,819 33. 118.019, 1.98, 1 .65Vo 35.0,2 37. p: 100/30 39. 76%o,2.3Vo (see Fig. 520 tn Sec. 24.8) Problem Set 25.2, page 1O48 3. l : pkT - p)n-k, $ : k/n,ft : number of successes in n trials 5. IIl20 7.1 : /(x), ó(ln )lap : Ilp - (x - t)l(I 9.[r:* 13.a:I Problem Set 25.3, page 1057 1. CONF6.g5 {37.47 5 p 1 43.87} 5.4,16 A57 -P):O,fi:llx 11. 0 : n/žxi: Ilí 15. Variability larger than perhaps expected 3. Shorter by a facto, XE 7. Cf . Example 2. n : 166 9, CoNFo gg{20.07 í P a 20.33} 11. CoNFo 99{63.71 a rl í 66.29} 13.c: 1.96,I:87, s2 :8J.413/5OO: J1,86,k: csl\/i: O.J43, CONF6.95 {86 < Ái = 88}, CONFo.gs{O.17

6019; do not reject the hypothesis. 5. ďln : l, c : 28.36; do not reject the hypothesis. 7. p < 28.76 or p ) 3I.24 9. Alternative p, + 1000, t: \/ň096 - 1000)/5 : _3.58 1c: _ 2.0g (Table A9, 19 degrees of freedom). Reject the hypothesis ,u : 1000 g. 11. Test LL:0 against p" + 0.t:2]1 < c:2.36 (7 degrees of freedom). Do not reject the hypothesis. 13. a: 5%o, c : 16.92 > 9. 0.52/0,42 : t4.06; do not reject hypothesis. t5. to: \1ď:9.I]lD (21.8 - 20.D/ffi : á.i; , : l:74 (17 degrees of freedom). Reject the hypothesis and assert that B is better. A58 App. 2 Answers to Odd-Numbered Problems 17.ug: 50/30 : I.6'7 1c:2.59 |(9,15) degrees of freedom]; do notrejectthe hypothesis. Problem Set 25.5, page 1071 1. LCL - 1 - 2.58 - 0.03l\/6: 0.968, UCL : L032 3.n:l0 5. Choose 4 times the original sample size (why?). l. z.sa\/0.0utÝž : 0.283,UCL : 2J.J83,LCL : 27.2I7 11. In 30Vo (57o) of the cases, approximately 13. UCL : np + 3\/"pO - D, CL : np,LCL: np - 3\/npa - e) 15. CL : LL:2.5,IJCL: p + 3\,G:7.Z,LCL: p - 3\/iis negative in (b) and we set LCL : 0. Problem Set 25.6, page 1O76 t. 0.9825, 0.9384,0.4060 3. 0.B181, 0.6703, 0.1353 5. P(A; 0) : e-3o0(t + loo) 7. P(A; 0) : e-io' g. I9.5vo, I4.]vo 11. (1 - 0)u, (l - 0)5 + 50(1 - 0)n 13. Because n is finite 15. o((9 - 12 + !1tÝt1l - on1):0,22 (if c : 9) 17. (I - ál' + 3.}rt - i)' : * Problem Set 25.7, page 1079 1.Xo': (30 - 50)2/50 + (70 - 50)2/50 : 16> c- 3.B4; no 3.41 5. Xoz :2.33 < c - 11.07. yes 7. rj: npj:37015 : J4, Xo2 :984114: I3.3, c:9.49. Reject the hypothesis. 9. Xo'- 1 < 3.84; yes 13. Combining the results forx : 10, 11, 12, wehave K - r - 1 : 9 (r : l since we estimated the mean,'r'# : 3.87). Xo2 : I2.9B 1 c : 16.92. Do not reject. 15. Xo' : 49/20 + 49160:3.27 1c :3.84 (1 degree of freedom, d: 57o),which supports the claim. 17.42 even digits, accept. Problem Set 25.8, pa8e l082 s. (})"(t + 18 + 153 + 816) : 0.0038 5. Hypothesis: Á and B are equally good. Then the probability of at least 7 trials favorable to A is }* + S,iu : 3.57o. Reject the hypothesis. 7. Hypothesis p : 0. Alternative pt } 0, í : 1.58, t : ÝIO, 1.58/1 .23 : 4.06 > c : L83 (a : Svo).Hypothesis rejected. 9.x:9.6],s: 1I.87,to:9.6]/(II.87l\/15) : 3.15 > c: 1,76(a: 57o). Hypothesis rejected. 11. Consider lj : xi - Fo. 13. P(T = 2) : 0.I%o from Table A12. Reject. ._+= _ App. 2 Answers to Odd-Numbered Problems 15. P(T = 15) : I0.8%a. Do not reject. 17. P(T = 2) : 2.8%o. Reject. Problem Set 25.9, page 109l l.y: 1.9 * x 3. y : 6.7407 + 3.068x 5. y : 4 + 4.8x, I72 ft 7.y: -1146 + 4.32x 9.y:0.5932 + 0.1138x,R: 1/0.1138 l1. qo : 76, K : 2.36\/16la . 944) : 0.253,CoNFo.95{-1.58 5 rr < -1.06} 13.3s*2: 500, 3s*u:33.5, kr : 0.06'7,3sr2 :2.268, Qo:0.023, K:0.02I CONF6.95{0.046 S, I4.5; reject p6. 23.It will double. 27. CONFo.g5{0.726 5 p a 0.751 } 31. CONF'.99{0.05 š cr2 < l0} _ l t4.]4 - 14.40 \ 35.0l _ |:0.9842 \ V0.025 l 37.30.I4l3.8 : "7.93 < 8.25. Reiect. 39. uo : 2.5 < 6.0 rQ,4) degrees of freedom]; accept the hypothesis. 41. Decrease by a factor Ž.gy a factor 2.5811.96 : I.32. 45.y: I.70 + 0.55x43. 0.9953, 0.9825, 0.9384, etc. APPENDlX Auxitiary Material A3.] Formulas for Special Functions 1 (4) Inx: , For tables of numeric values, see Appendix 5. Exponential functiotl e* (Fie. 5aa) (1) e : 2,J 1828 18284 59045 23536 0281411353 e*eU : e'*U, e*lea : e*-a, (e")' : e*U Natural logarithm (Fig. 5a5) (2) ln(xy):lnx + lny, In(xly):Inx - lny, ln(x") : a\nx lnx is the inverse of e', and elní : a, r-Inr : elnGlr) : 11r. Logarithm of base ten loglox or simply 1og x (3) Iogx: Mlnx, M: Ioge:0.43429 4481903251 82165 II289 I89I1 1og x, : 2.30258 50929 94045 68401 79914 54684 1og x is the inverse of 10', and 10l"g t : a,10-1og * : ílx. Sine and cosine functions (Figs. 546, 547).In calculus, angles are measured in radians, so that sin x and cos .r have period 2rr. sinx is odd, sin (-x) : -sinx, and cosx is even, cos (-x) : cos.x. -2O2x Fig.544. ExponentiaI function e' Fig. 545. Natura1 logarithm ln x 1 -:ln10M A60 A61 SEC. A3.1 Formu[as for Special Functions Fig. 546. sin x (8) (9) (10) (11) FQ.547. cos x sin 6 tan 6 cos 6 sin 6 tan 6 cos 6 1o : 0.01145 32925 19943 radian 1 radian'r'r r'n'r,r:;i::; sin2x * cos2 x: l I sin (x + y; : sin.T cos y * cos x sin y l ,in(x - y) : sinxcosy - cosx siny ) ] .o.(x + y) : cos-rcosy - sinx siny l I cos (* - y): cosJcosy f sinx siny sin 2x : 2 sin -tr cos J, cos 2x : cos2 x - sin2 x í sinx: .", (" -;): cos (+ - ,) t cosx : sin (" - +) :,* (+-' sin (zr - x) : sin x, cos (zr - x) : -cos _r cos2x : }(t l cos}x), sin2x : }(t - cos2x) I sin x sin y : +|- cos (x * y) * cos (x - y)] l,j cos _r cos y : }1cos (x + y) -t- cos (x - y)] I I sin _r cos y : };sin (x + y) f sin (x - y)] r ulu u-l) I Srnr/+slnU:2stn , cos , l I ulu u-t) ] cositf cosu:2cos , cos , l l ^ ulu u-l) I cosu-cosu:2sin-sin- L-22 (5) (6) (7) (I2) _BŤ A A! -B (13) Á cos x 1- B sin;r: \/A' + B' cos(x t 6), (14) Á cos x * B sinx: \/F+ B' sin(x t 6), A62 APP.3 AuxiliaryMaterial Fig. 5,48. tan x Tangent, cotangent, secant, cosecant (Figs, 548,549) sin x (l5) tanx: cos -r (16) (I]) ( 18) (19) (20) (2I) cos -rr cot -;r : --l-- , Sln -r sinhx : i@" - e-*), sinh x tanhx: cosh * ' cosh x * sinh x : er, cos -r tan(x-}): 1 cSCr: . Sln -r tanx - tan_y 1 SeC-r: - , tanx f tany tan(x i v):lull\Jl ')/ l-tanxtany 1 -l tan xtany Hyperbolic functions (hyperbolic sine sinhx, etc.; Figs. 550, 551) cosh x : i@" -l e-") cosh x COthX: sinh x coshx - sinh x: e-* cosh2x - sinh2 x: I sinhzx: }(cosh2x - I), cosh2x: }(cosh2x + I) Fig. 550. sinh x (dashed) and cosh x Fig. 55l. tanh x (dashed) and coth x Fig. 549. cot x SEC. A3.1 Formulas for Special Functions (:22) (23) (24) sinhx coshy + coshx sinhy coshx coshy + sinhx sinhy I sinh (x I cosh 1* ty): ty): A63 (a>0) From (24) we readily have f(1) : t; repeated application of (25) we obtain + 1) : af(a). hence if a is a positive integer, say k, then by tanhx + tanhy f(a+k+I) a(a-| 1)(a + 2) (a+k) tanh (x t y) : 1 -r tanh x tanhy Gamma function (Fig. 552 and Table A2 tnApp. 5). The gamma function f(a) is defined by the integral ^a:ó l fla; : l e_l ta_l clt Jo which is meaningful only if a > 0 (or, if we consider complex a, for those a whose real Part is Positive). Integration by parts gives the importantfunctional relation of the *amma function, (25) f(o (26) f(k+1):k! (k:0, 1,...). This shows that the gammafunction can be regarded as a Teneralization of the elementary factorial function. fSometimes the notation (a - 1)! is used for f(a), even for noninteger values of a, and the gamma function is also known as the factorial function.] By repeated application of (25) we obtain T-, f(a*1) f(a1-2) .\*,r: :... : a a(a+I) Fig. 552. Gamma function A64 APP. 3 Auxiliary Material and we may use this relation for defining the gamma function for negative (l (+ -I, -2, , , ,), choosing for k the smallest integer such that a -l k + 1 > 0. Together with (24), this then gives a definition oíl(a) for all a not equal to zero or a negative integer (Fig. 552). It can be shown that the gamma function may also be represented as the limit of a product, namely, by the formula (2]) (28) (3 1) (32) Beta function (33) (34) f(a + I) : \/-2,o (:)" From (27) or (28) we see that, for complex (x, the gamma function f(a) is a meromorphic function with simple poles at at:0, -1, -2,, , , An approximation of the gamma function for large positive a is given by the Stirling formula (29) where e is the base of the natural logarithm. We finally mention the special value (30) f1}1 : {n. Incomplete gamma functions ^íl P|a, x) : I e-ttu-r dt, Q(a, x) : Jo f(a) : P(a, x) -l Q@, x) i r-.'t'-' at (a > 0) Representation in terms of gamma functions: B(x, y) : tr--'(I - t)u-t 4, (x)O,y>0) Error function (Fig. 553 and Table 'A.4 in App. 5) (35) (36) erfx: + |'n-r'rit Yr Jo 2l,r3x5x7\ \/rr \ l!3 2|5 3|1 l (a * 0, -1, -2,, , ,) SEC. A3.1 Formulas for Special Functions Fig. 553. Error function erf (m; : I, complementary error function (37) erfcx - 1 - erfx: Fresnel integrals1 (Fig. 55a) A65 2r,,[; J-'-" dt (38) (39) C(*) : \/;l8, S(oo) : Ýnl8, complementary functions C(x) : /".o, 1t27 dt, S(x) : fri, 1t21 clt r_ :x) c(x; : l! - Ctxl:Icos(t2)clí v8 * r; rn s(x) : i - Slx;:J sin(tr)dt Y8 , r' sin t Si1x;: Jn , d' Sine integral (Fig. 555 and Table A4 in App. 5) (40) ,'s(r) ^z Fig. 554. Fresnel integrals 1Aucusttx FRESNEL (1188-1821), French physicist and mathematician. For tables see Ref. tGR1]. /' z-\ \, _ _/'_\j-a _-_J_ A66 APP. 3 Auxiliary Material Si(r) 2 n12 1 Silm; : 7rl2, complementary function (4I) si(x) Cosine integral (Table A4 in App. 5) (42) Exponential integral (43) Logarithmic integral (44) ci(,r) Ei(x) li(x) :I:l) Ťr 2 :ť :ť :r',o Si(x) sin r _dt t cos / _dt t e*t _dt t dt ln/ (x>0) (x>0) other notations are or by and z*(x1, yr); these may be used when subscripts are not used for another purpose and there is no danger of confusion. afI il |,,,.r,' í,(xr, yr) a, l I arl'í:t1,!1) A3.2 Partial Derivatives For dffirentiation formulas, see inside of front cover. Letz: f(x,})be areal functionof twoindependentrealvariables,xandy. If wekeep y constant, say, ! : !t, and think of x as a variable, then í(x, y) depends on _r alone. If the derivative of í(x,y) with respect to x for a value x : x1 exists, then the value of this derivative is called the partial derivative of í(x, y) with respect to x at the point (xr, yl) and is denoted by SEC. A3.2 Partial Derivatives A67 We thus have, by the definition of the rderivative, (l) af l .. f(xt*Ax,}r)- f(xt,yt)-: l :lim dx l,rr,r,, Ar-o Ax The partial derivative of z : f (x,y) with respect to y is defined similarly; we now keep -r constant, say, equal to x1, and differentiate f (xt, with respect to y. Thus "| :hm 6U l r*r,rr, Ay-o f (xt, yl + Ay) - f@t, y) Other notations arc f o(x1, y1) and zr(xt, !). It is clear that the values of those two partial derivatives will in general depend on the point (xr, yr).Hence the partial derivatives ózlóx and 6zlóy at a váriable point (x,y) are functions of ,r and y. The function Oz/6x is obtained as in ordinary calculus by differentiating z : f (x, y) with respect to x, treating y as a constant, and, dzldy is obtained by differentiating z with respect to y, treating x as a constant. EXAMPLE l Let7 : .f(x,y) : r'yi rsiny. Then 2xy -| sin 1, : *2 + Jcosy. The partial derivatives 6zl6x and 6zl8y of a function z : f(x, y) have a very simple geometric interpretation. The function z : f(x, y) can be represented by a surface in SPace. The equation Y : y1 then represents a vertical plane intersecting the surface in a curve, and the partial derivative ózlOx at apoint (xt,y) is the slope of the tangent (that is, tan a where a is the angle shown in Fig. 556) to the curve. Similarly, tňe partial derivative \z/óY at (x1, }r) is the slope of the tangent to the curve x : xl on the surface z : f (x, y) at (x1, y). Fig. 556. Geometrical interpretation of first partial derivatives Ay (2) aÍ l :\-, 0Y |,*,.r,, - l aí óy aí_ 6x A68 APP. 3 Auxiliary Material The partial derivatives ózlOx and 0zlEy are called first partial derivcttives or pclrtial derivatives of first order. By differentiating these derivatives once more, we obtain the folr second partial derivatives (or partial derivatives of second order)2 (3) a'í -,,dX' a'í ax óy a'f 0y 3x a'f _, ., dy, It can be shown that if all the derivatives concerned are continuous, then the two mixed partial derivatives are equal, so that the order of differentiation does not matter (see Ref. tGR4] in App. 1), that is, : * (#) :f" : * (#) :ío, : * (#) :f*, : * (#) :f,o ^, d,Z. 0x 0y ^, d-z :- 6y dx (4) EXAMPLE 2 ForthefunctioninExample 1. í**: 2y, í*u: 2x f cos ! : íg*, íuu: -x siny. l By differentiating the second partial derivatives again with respect to x and y, respectively, we obtain the third partial derivatives or partial derivatíves of the third order of /, etc. If we consider a function í(x,y, z) of three independent variables, then we have the three first partial derivatives f *(x, y, z), ír(*, y, z), and f ,(x, y, z).Here f * is obtained by differentiattng f with respect to x, treating both y and z as constalrís. Thus, analogous to (1), we now have etc. By differentiating f *, ío,f . again in this derivatives of /, etc. EXAMPLE 3 Letf(x,y,z): r'+ y'+ z2 + xy e'.Then ír:2xlyr', fu:2y*xez, írr:2, Í*u: ÍEa: ez, ír, : 2, íg": í"y: x z, f (xt + Ax, yt, z) - í(xr, yt,, z;.) L,x ' fashion we obtain the second partial í": 2z * xy ez, Ír": Ír*: Y e', í"": 2 l xy ez. I af 6x | :',* Ir*r,yr,rrl .rr -O 2 CAUTIOX! In the subscript notation the subscripts are written in the order in which we difí'erentiate, whereas in the "d" notation the order is opposite. SEC. A3.3 Sequences and Series A69 A3.3 Sequences and Series THEoREM t PRooF See also Chap. 15. Monotone Real Sequences We call areal sequence x1, x2, . increasing, that is, , xrL, . . . a monotone sequence if it is either monotone X1,5xz5xzš,.. or monotone decreasing, that is, xtŽ xzž ísž , , ,. We call X:', X21 " ' a bounded sequence if there is a positive constant K such that|xn| < rfor all n. Let x1, X2, ' ' ' be a bounded monotone increasing sequence. Then its terms are smaller than some number B and,, since \a xnfor alln, they lie in the intervalJr S xna B, which will be denoted bY 1o. We bisect Ig; that is, we subdivide it into two parts of equal length, If the right half (together with its endpoints) contains terms of the sequence, we denote it bY 1r. If it does not contain terms of the sequence, then the left half of 1o (together with its endpoints) is called !. This is the first step. In the second steP we bisect 1r, select one half by the same rule, and call it 1r, and so on (see Fig. 557 on p. A70). In this waY We obtain shorter and shorter intervals Ig, 11, Iz, . .. with the following ProPerties. Each lrncontains all Irfor n } m. No term of theiequence lies to the righi of lrn, and, since the sequence is monotone increasing, all x,-with n greater than some number N lie in In; of course, 1/ will depend on m, in general. The lengths of the I,n aPProach Zero aS m al!íoaches infinity. Hence there is precisely one number, call it L, that lies in all those intervals,3 and we may now easily p.ou" that the sequence is convergent with the limit Z. In fact, given an e } 0, we choose an m such that the length of lrris less than e. Then L and all the xn with n > N(m) lie in I,,, and, therefore, l;. - L:I'' a e for all those n. This comPletes the Proof for an increasing sequence. For a deóreasing sequence the proof is the Same, excePt for a suitable interchange of "left" and "right" iň tne construction of those intervals. l 3-, , - lnrs Statement Seems to be obvious, but actually it is not; it may be regarded as an axiom of the real number SYStem in the following form. Let J1, J2, , , ,be closed intervals such that each Jrrcontains all J.,wíth n ) m, and the lengths of the Jrn aPProach Zero aS m approaches infinity. Then there is precisely one real number that is contained in all those intervals. This is the so-called Cantor-Dedekind axiom, named after the German mathematicians GEoRG CANTOR (1B45-19lB), the creator of set theory, and RICHARD DEDEKIND(1B31-1916), known for his fundamental work in number theory. For further details see Ref. [GR2] in App. 1. (An interval 1is said to be closed if its two endpoints are regardád as points belonging to 1. lt is said to be openif the endpoints are not regarded as points of 1.) If a real sequence is bounded and monotone, it converges. A70 APP.3 Auxiliary Material ?_* I, Jr.J x2 Fig. 557. Proof of Theorem ] Real Series p R o O F Let sn be the nth partial sum of the series. Then, because of (1a), ,-1: í1, 3:2*x3Žs2, so that sz šsa Ésr. Proceeding in this fashion, we conclude that (Fig. 558) 52: Xt - X2 a S1', s3 : 1 - (xz - ís) š s1, Iim s2n: , *. n,+aa (3) which shows that the odd partial sums form a bounded monotone sequence, and so do the even partial sums. Hence, by Theorem 1, both sequences converge, say, lim,2r.11 :S, 7L+aJ 4 53 s1 Fig. 558. Proof of the Leibniz test Leibniz Test for Real Series Let x1, x2, , , , be real and monotone decreasing to Zero, that is, (1) (a) ír3xzžxsž",, (b) Iimx,n:0. Then the series with terms of alternating signs xt-xzixs-xs*-", conver7es, and for the remainder Rn after the nth term we have the estimate (2) ln,l = xn+l. s- s.JI THEoREM 2 SEC. A3.4 Grad, Div, Curl, V2 in Curvilinear Coordinates A71 Now, since s2n+I - s2n: x2n+1,, we readily see that (lb) implies ,' - s* : Ij-Szn+l -Jg 'r, : }!.(szn+l - szn):J]x x2n+I : 0. Hence s{' : .r, and the series converges with the sum .. We Prove the estimate (2) for the remainder. Since n 9 s, it follows fiom (3) that S2n+lŽ S Ž Sz, andalso S2n_lž, ž rrrr. By subtracting srn and s2n_7, respectively, we obtain S2n+1 - S2n ž 5 - Sznž 0, 0 > S - S2,-_IŽ Sz,' - Szn_I. In these inequalities, the first expression is equal to x2n_yl,the last is equal to -x2n, aí7d the exPressions between the inequality signs are the remainder s Rz, and R2,_1. rňus the inequalities may be written xzn+lžRr.,-ž 0, 0ž Rzr_tž -xzn and we see that they imply (2). This completes the proof. l in A3.4 Grad, Div, Cur[, V2 cu rvi l inear coord i nates To simplify formulas we write Cartesian coordinates í : xL, j : x2, Z : x3. We denote curvilinear coordinates bY Qt, Qz, q3. Through each point F in"." iass three coordinate surfaces Q1 : CohSt, Q2 : Const, q3 : consr. They intersect along coordinate curves. We assume the three coordinate curves through P to be orthogonul 1p..p"ndicular to each other). We write coordinate transformations as (1) Xt : Xl(Qt, Qz, cls), xz: xz(Qí, q2, q3), xs : xs(Qt, q2, q3). CorresPonding transformations of grad, div, curl, and V2 can all be written by using (2) hj,:Ž (t)' Next to Cartesian coordinates, most important are cylindrical coordinates 4l : T, Qz: 0, Qs : z (Fig. 559a on p. A12) defined by (3) xt : Ql cos q2: r cos 0, x2 : qlSinQz: rsin0, x3: Q3: Z and spherical coordinates QI : l", Qz: 0, Qs : ó @ig. 559b) defined bya (4) Xl: Ql,Cosq2 sinq3: rcos 0sinQ, xz: Qtsinqrsin43: rsin 0sin@ x3: Ql cos q3 : r cos @. aThi' i* the notation used in calculus and in many other books. It is logical since in it, 0 ptays the same role as in polar coordinates. CAUTION! Some books interchange the roles of 0 and $. A72 APP. 3 Auxiliary Material Gradient. 4-system, coordinate g (r,0, z) 1 l I lz I '|----------- y (r, O,Q) (o) cylindrical coordinates (ó) spherical coordinates Fig. 559. Special curvilinear coordinates In addition to the general formulas for any orthogonal coordinatas Qt, Qz, Q3, we shall give additional formulas for these important special cases, Linear Element ds. In Cartesian coordinates, ds2:dxrz+dxr2+dx"2 (Sec. 9.5). For the q-coordinates, (5) ds2 (5) For polar coordinates set clz2 : O. () .) ds2 : drz + ,2 sin2 ó ,l02 + r2 d62 (Spherical coordinates). |-----____- hr' d.q * hr' dqr' * h"' cJqz2. tls2 : drz + ,2 ,l02 + dz2 (Cylindrical coordinates), grad f : V/ : lf*r, f*r, í*") (partial derivatives; Sec. 9.7). In the with u, v, w denoting unit vectors in the positive directions of the Qr Qz, Qz curveS, respectively, (6) gradf : Ví (6') grad / : Ví: (6") graď f : V/ : Divergence div F - V,F - 1 (]) divF:V.F: hrhrh" (,7') divF : V.F : !*r"orl rdr laIlaf -r--!-i hl óqt h2 dqz af l a.f aí -u + -dr ró0 óz af 1 óí l aí -t---}r- _ll dr rsinQ á0 r aó 1af h3 óqs 9,8); ",] Fz, Fs), a +_óqs , aFz T 0z (Fr)", t Tól_ l_ L dqt 1 óFz l_' , d0 (F)*, -| (Fs),, (F : [Fr, ó (h2hF) * w (hhlFz) (Cylindrical coordinates) (Spherical coordinates). Sec. (hrhz (Cylindrical coordinates) _+ _ 0 '., SEc. A3.4 Grad, Div, Curl, V2 in Curvilinear Coordinates (]") divF : V.F : +r 6,r^ | }Fz l ó -(r-F, } f .ór'" I rsin$ a0 *.rró--(sinÓFs| Laplacian Y'f : V.V,f : div (grad í): f *rrr+ f *r,, + (8) Y,f:#l+(T#)-*(+ (B,) y,f :! r! * +I 4rlr- r dr r- Í)0- áZ.. (8,,) Y2r:Ů_*?!Lr. ' a'f l a'í ,2tó aí 'J ar'-;íh'',; o*r*7aď* ,, oW Curl (Sec. 9.9): A73 (Spherical coordinates). (Cylindrical coordinates) (Spherical coordinates). .frr". (Sec. 9.8): #)-*(T #)] (9) curlF:VxF: 1 hlh2h3 hta a 0qt htFt hzv hsw dd óqz 6qz hzFz hsFs For cylindrical coordinates we have in (9) (as in the previous formulas) h1: hr: l, hz: ha: 4t : ť, h" : hr: 7 and for spherical coordinates we have h1 : h,: 1, hz: hr: q1 sin qr: r sin $, he: ha: Qt APPENDlX is identically zero on 1,, then lt = lz on 1, which implies uniqueness. Since (1) is homogeneous and linear, y is a solution of that ODE on I and since y1 and y2 satisfy the same initial conditions, y satisfies the conditions y(.ro) : 0, y'(-ro) : 0. z(x): y(x)z + y'(*)' z' :2yy' l 2y'y". y :-py -qy. the expression for z' ," obtain z':2yy'-2py''-Zqyy' By substituting this in (11) Now, since y and y real, (y t y')' : y2 ! 2yy' + y'' = O. 1This proof was suggested by my colleague, Prof. A. D. Ziebur. In this proof we use formula numbers that have not yet been used in Sec. 2.6. Additional proofs Section 2.6, page 73 PROOF OF THEOREM 1 Uniqueness1 Assuming that the problem consisting of the (l) and the two (3) has two solutions difference (10) we consider the function and its derivative From the ODE we have y" + p(x)y' oDE -l q(x)y : 0 initial conditions y(xo) : Ko, y'(xo) : Kt !{x) and y2@) on the interval 1 in the theorem, we show that their y(x):yr(x)-yz@) A74 | *_ APP.4 Additional Proofs (I2) (13a) (I4) From this and the definition of z we obtain the two inequalities (a) 2yy' = y2 + y'' : z. (b) -2yy' 5 y' + y'' : z. A75 for all x on I. From (I2b) we have 2yy' > -z. Together,|2yy'l šz. For the last term in (11) we now obtain -2qyy' = 1-2qyy' | : lqll2yy' | = lqlz. Using this result as well as -p < |p| and applying (12a) to the term2yy' in (11), we find z'az+2lp|y'r+lqlr. Since y'' = y2 + y'' : z, from this we obtain z'=(I+2lpl+lď, or, denoting the function in parentheses by ň, z'=hz Similarly, from (11) and (l2) it follows that (13b) -a' : -2yy' -l 2py'' 1- 2qyy' =z*zlplz+lqlz:hz. The inequalities (13a) and (13b) are equivalent to the inequalities z' - hz=O, z' + hz> 0, Integrating factors for the two expressions on the left are Ft : ,-Ih@) dr and Fz : níll,(rl dr. The integrals in the exponents exist because ň is continuous. Since F1 and F2 arepositive, we thus have from (14) Fr(z'-hz):1Frz)'<0 and Fzk'+hz):(F2z)'>0. This means that F3 is nonincreasing and F2z is nondecreasing on 1. Since z(xg) : 0 by (10), when x š.T6 we thus obtain F3 ž (Flz)*o: 0, and similarly, when x ž xg, Flz S 0, FqZš(F"Z,\- -0- , ,^í) F2z > 0. Dividing bY F, and F, and noting that these functions are positive, we altogether have zš0, z>0 for all x on L Thisimpliesthat z: y2 l y'' = 0on1. Hencey = 0of .}r = yronI. l A76 APP.4 Additional Proofs Section 5.4, pages ]84 p R o o F o F T H E o R E M 7 Frobenius Method. Basis of Solutions. Three Cases The formula numbers in this proof are the same as in the text of Sec. 5.4. An additional formula not appearing in Sec. 5.4 will be called (A) (see below). The ODE in Theorem 2 is (l) y"+b(x) rt++y:0,X'X' where b(x) and c(x) are analytic functions. We can write it (1') ,'y" + xb(x)y' * c(x)y : 0. The indicial equation of (1) is (4) r(r - I) -f bgr-l c6 : 0. The roots ť1, T2of this quadratic equation determine the general form of a basis of solutions of (1), and there are three possible cases as follows. Case 1. Distinct Roots not Differing by an Integer. A first solution of (1) is of the form (5) and can be determined as in the power series method. For a proof that in this case, the ODE (1) has a second independent solution of the form !z(x) : x'r(Ao l Alx l A2x2 + . . .), yr(x) : x'r(ao -| alx t arxz + . . .7 (6) (1) see Ref. tAl1] listed in App. 1. Case 2. Double Root. The indicial equation (4) has a double root r if and only if (bo- l)'- 4co:0, andthenr:Ž(t - b . Afirstsolution yr(x) : x' (ao l alx l a2x2 +,,,), r:lG-bo), can be determined as in Case 1. We show that a second independent solution is of the form (8) yz@): h(x) lnx * x'(Alx l A2x2 + " ,) (x > 0). We use the method of reduction of order (see Sec. 2.I),that is, we determine u(x) such that yz@) : u(x)y{x) is a solution of (1). By inserting this and the derivatives yL : u'y1 + uy'1, y'J, : u"y, * 2u'y',,, + ,y'], into the ODE ( 1 ') we obtain x2(u"yl + 2u'y'1 + ,y'l) + xb(u'y1 + uyb l cuy1 :0. APP.4 Additional Proofs (9) Since Y1 is a solution of (1'), the sum of the terms involving u is zero, and this equation reduces to *'yru" + 2x2y'tu' + xbyrLt' : O. By dividingby xzyt and inserting the power series for b we obtain 'yibn\,,t"+(r- + - +...lr,,:0. \ y, x l- V' Here and in the following the dots designate terms that are constant or involve positive powers of x. Now from (7) if follows that '', : x'-lfrao - ? 1- I)arx * "] }r x'lao* alx +...] x \ ao*alx +... r -+....x Hence the previous equation can be written L,,,+(":+- )Lt,:0.\,/ Since r: (I - b 12, the term (2r + bg)lx equals llx, andby dividingby u'we thus have #:-+BY integration we obtain Inu' : -lnx + ..., hence u, : ('lx)e( "). Expanding the exPonential function in Powers of x anď integrating once more, we see thatu is of the form u: lnx l kp l krxz + . . .. Inserting this into lz: ujt, we obtain for y2a representation of the form (8). Case 3. Roots Differing by an Integer. We write T1_: T and rr: r - p where p is a positive integer. A first solution l,t,, + (r, * uo * .. \x ): yr(x) : x'r(ao - alx l arx2 + . . .1 can be determined as in cases 1 and 2. we show that a second independent solution is of the form (10) +...) : u!t.The first steps are yz@) : kyl@) ln x f x',(Ao -l Ap l A2x2 where we may have k * 0 or k : 0. As in Case 2 we set y2 literally as in Case 2 and give Eq. (A), ) tt' : 0. A78 APP.4 AdditionalProofs Now by elementary algebra, the coefficient bg - 1 of r in (4) equals minus the sum of the roots, bo - I : -(rt,l rr) : -(r l r - p) : -2r -l p. Hence 2r -l bg: p + 1, and division by u' gives The further steps are as in Case 2.Integtating, we find lnu':-(p+ 1)lnx+ ., thus t -(p+ 1) (. . .) u:X e where dots stand for some series of nonnegative integer powers of x. By expanding the exponential function as before we obtain a series of the form ?:-(+- ) ,,,_ l -k' l... _rkr- ' k^ u: xel* ť +",+Ž+--J!-lkp.rlko+zx +",. /tk u:kolnxi Í-- _ 'p-L ,-i,\p*ox Hence, by (9) we get for y2: ult the formula / ,r:koltln,r* *"-o ( ; -ko_txo-I But this is of the form (10) with k: ko since \ - p series involves nonnegative integer powers of x only. Section 5.7, page 205 We integrate once more. Writing the resulting logarithmic term first, we get lko+ - ) \ +",l(ao+agl",). l : f2 zfld the product of the two THEoREM Reality of Eigenvalues Iíp, q, r, and p' in the Sturm-Liouville equation (I) of Sec. 5.'7 are real-valued and continuous on the interval a š x = b and r(x) > 0 throughout that interval (or r(x) < O throughout that interval), then all the eigenvalues of the Sturm-Liouville problem (I), (2), Sec. 5.J, are real. PROOF Let.tr: a* iBbe aneigenvalueof theproblemandlet y(x) -- u(x) + iu(x) be a corresponding eigenfunction; here d, F, u, andu are real. Substituting this into (1), Sec. 5.7, we have (pr' t ipr')' + (q l ar + iBr)(u * iu) : 0. APP.4 Additional Proofs This complex equation is equivalent to the imaginary parts: A79 the following pair of equations for the real and (pu')' + (q + ar)u - Bru :0 (pu')' + (q -l ar)u * Bru: O. Multiplying the first equation by u,the second by -, and adding, we get -Ffu2 + u2)r: u(pu')' - u(pu')' : |(pu'), - (pu')r]' . The expression in brackets is continuous on a šx a b,for reasons similar to those in the Proof of Theorem 1, Sec. 5.7. Integrating over -r from a to b, we thus obtain rb F lb -P J (u2 + uz)r dx : I p@r' - u,rl| Because of the boundary .oiOi,ion, the right ,rL, zero;,nr, ], ". in that proof. Since y is an eigenfunction, u2 + u2 * 0. Since y and r are continuous and, r > 0 (or r { 0) on the interval a šx S b, the integral on the left is not zero. Hence, B: 0, which means that ,[ : a is real. This completes the proof. l Section 7.7, page 308 THEoREM Determinants The definition of a determinant (1) D: detA: att azt atz azz aln a2n anl an2 ann as given in Sec. ].7 is unambiguous, that is, it yields the same value of D no matter which rows or columns we choose in developings. P R O O F In this Proof we shall use formula numbers not yet used in Sec. 7.7. we shall prove first that the same value is obtained no matter which row is chosen. The proof is by induction. The statement is true for a second-order determinant, for which the develoPments by the first row atlazz l ar2(-a21) and, by the second row azl(-an) * a22al give the same value attazz - atzazt. Assuming tňe statement to be true for an (n l)st-order determinant, we prove that it is true for an nth-order determinant. A80 APP.4 AdditionalProofs For this purpose we expand D in terms of each of two arbitrary rows, say, the ith and the jth, and compare the results. Without loss of generality let us assume i < j. First Expansion. We expand D by the ith row. A typical term in this expansion is aa"C* : aik' (- I)'*OMno. The minor Mu" of ap in D is an (n - 1)st-order determinant. By the induction hypothesis we may expand it by any row. We expand it by the row coíTesponding to the jth row of D. This row contains the entries a7(l * k).It is the (7 - 1)st row of Mp,because Mp does not contain entries of the ith row of D, and i < j. We have to distinguish between two cases as follows. Case I. If / < k, then the entry ailbelongs to the /th column of M6 (see Fig. 560). Hence the term involving ail ln this expansion is (2O) a1. (cofactor of ail If- M*) : an- tI)Q-D*'Mroj, where Ma"il is the minor of altn Mp. Since this minor is obtained from M*by deleting the row and column of an, it is obtained from D by deleting the ith and jth rows and the kth and /th columns of D, We insert the expansions of the M4, into that of D. Then it follows from (19) and (20) that the terms of the resulting representation of D are of the form (19) (2Ia) where (2lb) where ó is the same as before. ai1rail' (- l)'Mnoi, b : i + k + j + l - 1. -aipa1. (-I)bMa"it Case II. If l > k, the only difference is that then a3L belongs to the (/ - 1)st column of Mp,because M6 does not contain entries of the frth column of D, and k < /. This causes an additional minus sign in (20), and, instead of (2Ia), we therefore obtain (l k) lth kth col. col. ll lI |^____]_ Ja_.Y--_ tV/ ll lI --ío.L -l----\ulIl ll ll Case I lI || ,\| -Ja..l-----L---\ LhJ \_-/ l ||IA ---| -Ja L__IV/ |||||| Case II hth col. ith col. lth row jth row Fig. 56O. Cases l and ll of the two expansions of D APP.4 AdditionalProofs unambiguous. Section 9.3, page 377 PRooF oF FoRMULA (2) Second Expansion. We now expand D at first expansion is A8l by the jth row. A typical term in this (22) ailC1 : ajl'(-Dj*'Mir. By the induction hypothesis we may expand the minor Ml of a3lin D by its ith row, which conesponds to the ith row of D, since j > i. Case I. If k > l, the entry a6 tn that row belongs to the (k - 1)st column of Mn, because Mil does not contain entries of the /th column of D, and l { k (see Fig. 560). Hence the term involving a6 in this expansion is (23) a7". (cofactor of a6 In Mi) : aar. (- l)l+0suchthat We maY now take n so large that the triangle Tnlies in the disk lz - zol < á. Let Lnbe the length of Cn. Then l, - ,ol 1 Lnfor all z on Cn and, 79 in T,. From this and (4) we have!h(x)(z - z | 1 eLn.The ML-inequality in Seó. 14.1 now giu", lf.,,ra orl : If",,r*r*,zo) orl= Ln. Ln : Ln2. Now denote the length of C by Z,. Then the path C1 has the length Lt : LlZ, the path C2 has the \ength L, : L1l2 : Ll4, etc., and Cn has the leÁgth'L,: Ll2n. Hence Ln' : L2/4n. From (2) and, (5) we thus obtain |f., "l= o,1f",", o,Ia 4neLnz : o-, # A90 listed in App. 1. Fig. 563. Proof of Cauchy's integral theorem for a polygon Section l5.1, page 667 P RO O F O F T H E O R E M 4 Cauchy's Convergence Principle for Series (a) In this proof we need two concepts and a theorem, which we list first. APP.4 Additional Proofs By choosing e () 0) sufficiently small we can make the expression on the right as small as we please, while the expression on the left is the definite value of an integral. Consequently, this value must be zero, and the proof is complete. The proof for the case in which C is the boundary of a polygon follows from the previous proof by subdividing the polygon into triangles (Fig. 563). The integral corresponding to each such triangle is zero. The sum of these integrals is equal to the integral over C, because we integrate along each segment of subdivision in both directions, the corresponding integrals cancel out in pairs, and we are left with the integral over C. The case of a general simple closed path C can be reduced to the preceding one by inscribing in C a closed polygon P of chords, which approximates C "sufficiently accurately ," and it can be shown that there is a polygon P such that the integral over P differs from that over C by 1ess than any preassigned positive real number E, no matter how small. The details of this proof are somewhat involved and can be found in Ref. [D6] 1. A bounded sequence,1,,2: ,,,is a sequence whose terms all lie in a disk of (sufficiently large, finite) radius K with center at the origin; thus |s,| { K for aLI n. 2. Alimitpoint aof asequence.i1,,2, ",is apoint suchthat, given an e) 0, there convergence, since there may still be infinitely many terms that do not lie within that circle of radius e and center a.) has the limit points 0 and 1 and diverges. 3. A bounded sequence in the complex plane has at least one limit point. (Bolzano-Weierstrass theorem; proof below. Recall that "sequence" always mean infinite sequence.) converges if and only if for(b) We every e ) (1) now turn to the actual proof that zt a zz + 0 we can find an N such that Here, by the definition of partial sums, Sn,p Sn:Zr,t* -|in,p. lZr*t+,,, l zn*ol 1, for every n } N anď p : I, 2, "-1,1/ "_-'> t Y tr APP. 4 Additional Proofs A91 Writing n -| p: r)we see from this that (1) is equivalent to (1*) |s,-s,| N. SuPPose that s1, 2" " converges. Denote its limit by s. Then for a given e ) O we can find an N such that lle lS,-'rl <' Hence, if r> Nand ,;, N, then by the triangle inequality (Sec. I3.2), lr, - r| : l(s, - s,.) l (sn- r)l = lr, - r,"l * |r,, - s| < that is, the sequenc s1, s2, , , . is convergent with the limit s. lr,-r,"l :l(s,-s)- (sn-r)l =lr,-r| i|r,"-s| < that is, (1*) holds. (c) ConverselY, assume that s1, s2, ",satisfies (1*). We first prove that then the Sequence must be bounded. Indeed, choose a fixed e and a fixed ft: lto > 1/in (1*). Then (1*) imPlies that all s, with r > N lie in the disk of radius e and center srro and only finitelY manY terms 1, , , , , s^ may not lie in this disk. Clearly, we can now find a circle so large that this disk and these finitely many terms all lie within this new circle. Hence the sequence is bounded. By the Bolzano-Weierstrass theorem, it has at least one limit point, call it s. We now show that the Sequence is.convergent with the limit s. Let e ) 0 be given. Then there is an N* such that lr, - ,,rl < el2 for all r ) N* andn > ly'*, by (1*). Also, bY the definition of a limit Point, |r," - s| < e/2 for infinitely manyn, so that we can find and fix ann} N* such that |sr, - s| < el2.Together, for every. > N*, ' -l 2' 2 -e! foreveryn}N. :;. -+-22 THEoREM P R O O F It is obvious that we need both conditions: a finite sequence cannot have a limit point, and the Sequence I,2, 3,, , , , which is infinite but noi bounded, has no limit point. To Prove the theorem, consider a bounded infinite sequence zl, Z2,. . . and Iet Kbe such that |r.l < K for all n.If only finitely many values of the zn are different, then, since the sequence is infinite, some number z must occur infinitely many times in the sequence, and, bY definition, this number is a limit point of the ,"qu"n"". We maY now turn to the case when the sequence contains infinitely many dffirent terms. we draw alarge square Qothatcontains all7n.we subdivide Qginio four congruent Squares, which we number 1,2,3, 4. Clearly, at least one of these ,quu.", (each taken SgpRxARo BOLZANO (l781-1B48), Austrian mathematician and professor of religious studies, WaS a Pioneer in the studY of point sets, the íoundation of analysis, and mathematical logic. For Weierstrass, see Sec. l5.5. Bolzano-Weierstrass Theorem3 A bounded infinite sequence ZI, Z2, Z3, limit point. in the complex plane has at least one A92 APP.4 Additional Proofs with its complete boundary) must contain infinitely many terms of the sequence. The square of this type with the lowest number (I,2,3, or 4) will be denoted by Qt This is the first step. In the next step we subdivide Q, into four congruent squaíes and select a square Qz by the same rule, and so on. This yields an infinite sequence of squares Qo, Qt, Qz, . . , Qn,. . . with the property that the side of Qnapproaches zero as n appíoaches infinity, and Q* contains all Qn with n ) m. It is not difficult to see that the number which belongs to all these squares,4 call it z : a, is a limit point of the sequence. In fact, given an e ) 0, we can choose an N so large that the side of the square 0, is less than e and, since Qry contains infinitely many zn, we have |zn - a| < e for infinitely many n. This completes the proof. Section l5.3, pages 681-682 PART (b) oF THE PRooF oF THEoREM we have to show that 5 17+ Lz)" - zn _,r.-'f :ŽranLz[(. + Lz)n-2 + 2z(z* Lz)n-" + ", + (n- I)zn-'f, thus, 17+ Lz)" - zn - n7n-I : Lzlk+ L,z)n-2 +Zz(za L,z)n-" +... + (n- I)z'-'l. If we setz -| L,z: b andZ: a, thus Ae : b - a, thisbecomes simply ž,,"I L,z A,z (7a) bn-a' _ nan-7 : (fu - a)An b-a (n:2,3,, , ,), where An is the expression in the brackets on the right, (7b) An : bn-2 + 2abn-3 + 3a2bn-4 + . . . + (n - I)an-2; thus, Á2 : I, As: b + 2a, etc. We prove (7) by induction. When fr: 2, then (7) holds, since then b2-a2 - n^ -LLÁ - b-a (b+a)(b-a) _2a-b-a:(b-a)Az. b-a Assuming that(7) holds for n: k,we show that it holds for n: k + 1.By adding and subtracting a term in the numerator and then dividing we first obtain 5k+1, _ ok+l bk+I _ bak + 5ok _ ok+I b-a b-a bk-ak -b" "' +ak. b-a 4The fact that such a unique number z : a exists seems to be obvious, but it actually follows from an axiom of the real number system, the so-called Cantor-Dedekind axiom., see footnote 3 in App. A3.3. n APP.4 Additional Proofs A93 DirectBy the induction hypothesis, the right side equals u|ru - a)Al, -| koo-'\ + ak. calculation shows that this is equal to (b - a){bAo -| kak*'} + akak-r + ak. From (7b) with n : k we see that the expression in the braces {. . .} equals bk-r -l 2abk-2 + . . . + (k - I)bak-2 l kak-l - Au*r. Hence our result is bk+I _ ak+l b-a : (b - a)Arc*t + (k + I)ok. Taking the last term to the left, we obtain (7) with n : k * 1. This proves (7) for any integer n > 2 and completes the proof. l Section l8.2, page 754 ANOTHER PROOF OF THEOREM 1 without the use of a harmonic conjugate We show that if w : Ll l iu : í(z)is analytic and maps a domain D conformally onto a domain D* and Q*(u, u) is harmonic in D*, then (1) Q(x, y) : Q*(u(x, y), u(x, y)) is harmonic in D, that is, V2ó : 0 in D. We make no use of a harmonic conjugate of O*, but use straightforward differentiation. By the chain rule, ó" : Qu* ,* * ór* u*. We apply the chain rule again, underscoring the terms that will drop out when we form V2ó: Q**: Qu*u** -l (@juu* l Q*uu*)u* l Q,*u,* + (9j"!, + O*, u*)u*. Q* is the same with each x replaced by y. We form the sum V2O. In it, óf, : ó}, is multiplied by tt*U* * ttouo which is 0 by the Cauchy-Riemann equations. Also Y2u :0 and Y2u :0. There remains V2Q : Qť,u(u*' + ur2) -l Q!,(u*z + urz). By the Cauchy-Riemann equations this becomes V2o: (ať,.+ Q!,)(u,'* r,r) and is 0 since O* is harmonic. l Tables For Tables of Laplace transforms see Secs. 6.8 and 6.9. For Tables of Fourier transforms see Sec. 11.10. If you have a Computer Algebra System (CAS), you may not need the present tables, but you may still find them convenient from time to time. Table Al Bessel Functions For more extensive tables see Ref. tGRl] in App. 1. Jo(,r) : 0 for x : 2.40483, 5.52008, 8.65313, I|.1915, 14.9309, 1 8.071 1, 21.2116, 24.3525, 21 .4935, 30.6346 JlQ'1 :0forx:3.83171, 1.01559,l0.1735, 13.3237,16.4106,19.6159,22.7601,25.9037,29.0468,32.1897 -T Jo@) /.(x) X /o(x) Jr(x) x /o(x) /r(.;r) 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.1 0.8 0.9 1.0 1.1 |.2 1.3 1.4 1.5 1.6 I.1 1.8 I.9 2.0 2.I 2.2 2.3 2.4 2.5 2.6 2.1 2.8 2.9 1.0000 0.9975 0.9900 0.9776 0.9604 0.9385 0.9120 0.88l2 0.8463 0.8075 0.,7652 0.1196 0.671l 0.620I 0.5669 0.5118 0.4554 0.39B0 0.3400 0.2818 0.2239 0.1666 0.1104 0.0555 0.0025 -0.0484 -0.0968 -0.1424 -0.1850 -0.2243 0.0000 0.0499 0.0995 0. l 483 0.1 960 0.2423 0.2861 0.3290 0.36B8 0.4059 0.4401 0.4109 0.4983 0.5220 0.5419 0.5519 0.5699 0.57,18 0.5815 0.5812 0.5161 0.5683 0,5560 0.5399 0.5202 0.491l 0.4708 0.4416 0.4097 0.3154 3.0 3.1 3./- J.J 3.4 3.5 3.6 3.7 3.8 3.9 4.0 4.1 4.2 4.3 4.4 4.5 4.6 4.7 4.8 4.9 5.0 5.1 5.2 5.3 5.4 5.5 5.6 5.,7 5.8 5.9 -0.2601 -0.2921 -0.3202 -0.3443 -0.3643 -0.3B01 -0.3918 -0.3992 -0.4026 -0.40l8 -0.3971 -0.3887 -0,3,766 -0,3610 *0.3423 -0.3205 ,0,296I -0.2693 -0.2404 *0.2091 -0.1716 -0.1443 -0.1 103 -0.0758 -0.0412 -0.0068 0.0270 0.0599 0.0917 0.1220 0.3391 0.3009 0.2613 0.220] 0.1192 0.I374 0.0955 0.0538 0.0128 -0.0212 -0.0660 -0.1033 -0.1386 -0.I719 -0.2028 -0.23II -0.2566 -0.2191 -0.2985 -0.3141 0.3216 -0.3311 -0.3432 -0.3460 -0.3453 ,0.3414 -0.3343 -0.3241 -0.31 l0 -0.295I 6.0 6.1 6.2 6.3 6.4 6.5 6.6 6.1 6.8 6.9 1.0 1.1 ,7.2 l.J 1.4 7.5 1.6 1.1 1.8 1.9 8.0 8.1 8.2 8.3 8.4 8.5 8.6 8.1 8.8 B.9 0.1506 0.1113 0.2017 0.2238 0.2433 0.260I 0.2740 0.285l 0.293I 0.2981 0.3001 0.2991 0.z95I 0.2882 0.2186 0.2663 0.2516 0.2346 0.2154 0,1944 0.n11 0.1415 0.1222 0.0960 0.0692 0.0419 0.0146 -0.0125 -0.0392 -0.0653 -0.2767 -0.2559 -0.2329 -0.208l -0.1816 -0.1538 -0.1250 -0.0953 -0.0652 -0.0349 -0.0041 0.0252 0.0543 0,0826 0.1096 0.1352 0.1592 0.1813 0.2014 0.2192 0.2346 0.2476 0.2580 0.2651 0.2708 0.2731 0.2728 0.2691 0.2641 0.2559 A94 APPENDlX APP. 5 Tables A95 Table A2 Gamma Function [see (24) in App. A3.1] Table A4 Error Function, Sine and Cosine lntegrals [see (35), (40), (42) in App. A3.1] Table Al (continued) X Yo@) Yt@) x Yo@) Yt@) x Yo@) Yt@) 0.0 0.5 1.0 1.5 2.0 (-*) -0.445 0.088 0.382 0.510 (-o) - I.47l -0.781 ,0,412 -0.107 2.5 3.0 3.5 4.0 4.5 0.498 0,377 0.189 -0.017 -0.195 0.|46 0.325 0.410 0.398 0.301 5.0 5.5 6,0 6.5 7.0 -0.309 -0.339 -0.288 -0.113 -0.026 0.148 -0.024 -0.175 -o.214 -0.303 q f(o) a f(a) a f(a) a f(a) (y f(a) 1.00 I.02 1.04 1.06 1.0B 1.10 1.12 1.14 1.16 1.18 1.2o 1.000 000 0.988 844 0.978 438 0.968744 0.959 725 0.951 351 0,943 590 0.936 416 0.929 803 0.923128 0.918 169 1.20 1.22 1.24 I.26 I.28 1.30 1.32 I.34 1.36 1.3B I.40 0.918 169 0.913 106 0.908 521 0,904 397 0.900 718 0.897 47l 0.894 640 0,892216 0.890 185 0.888 537 0,887 264 1.40 1.42 1.44 1.46 1.48 1.50 1.52 I.54 1.56 1.58 1.60 0.887 264 0.886 356 0.885 805 0.885 604 0.885 747 0.886 227 0.887 039 0.888 178 0.889 639 0.891 420 0.893 515 1.60 1.62 1.64 1.66 1.68 I.70 1.12 I;74 1.16 1.78 1.80 0,893 515 0.895 924 0.898 642 0.901 668 0.905 001 0.908 639 0.912 58l 0.916 826 0.921375 0.926 227 0.931384 1.80 1.82 1.84 1.86 l.B8 1.90 1.92 1.94 1.96 1.98 2.00 0.931384 0.936 845 0.942 612 0.948 687 0.955 071 0.961766 0.968 7]4 0.976 099 0.983 743 0.991 708 1.000 000 Table A3 Factorial Function and lts Logarithm with Base lo n n! log (n!) n n]. log (n!) n nI log (n!) 1 2 _-) 4 5 1 2 6 24 120 0.000 000 0.301 030 0,77815I 1.380 21 1 2.079 t&l 6 7 B 9 10 720 5 040 40 320 362 880 3 628 800 2.857 332 3.102 431 4.605 52I 5.559163 6.559 763 11 l2 13 l4 l5 39 916 800 479 001 600 6227 020 800 81 l78291 200 l 301 674 368 000 7.601 156 8.680 337 9.794 280 10.940 408 12.116 500 X erf x Si(x) ci(x) x ertx Si(x) ci(;r) 0.0 0.2 0.4 0.6 0.8 1.0 I.2 1.4 L6 1.8 2.0 0.0000 0.2227 0.4284 0.6039 0.742l 0.8427 0.9103 0.9523 0.9163 0.9891 0.9953 0.0000 0.1996 0.3965 0.5881 0.7]2I 0.946I 1.1080 I.2562 1.3892 1.5058 1.6054 @ I.0422 0.3788 0.0223 -0.1983 -0.3374 -0.4205 -0.4620 -0.4,711 -0.4568 -0.4230 2.0 2.2 2.4 2.6 2.8 3.0 3.2 3.4 3.6 3.8 4.0 0.9953 0.998i 0,9993 0.9998 0.9999 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.6054 I.6876 I.1525 1.8004 I.832l 1.8487 1.8514 1.8419 1.8219 1.1934 1.7582 -0.4230 -0.3751 -0.3l73 -0.2533 -0.1865 -0,l 196 -0.0553 0.0045 0.0580 0.1038 0.1410 A96 APP. 5 Tables Probability function /(x) [see (2), Sec. 24.]l and distribution function F(x) n r p:0.| í(x)l or,l p-0.2 .f(.r) | orrl p:0.3 f(x) i .r.rl p:0.4 í(.r)| or"l p:0.5 í(r)] rrr) 1 0 1 0. 9000 1000 0.9000 1.0000 0. 8000 2000 0.8000 1.0000 0. 7000 3000 0.7000 1.0000 0. 6000 4000 0.6000 1.0000 0. 5000 5000 0.5000 1.0000 2 0 1 2 8 100 1 800 0100 0.8100 0.9900 1.0000 6400 3200 0400 0.6400 0.9600 1.0000 4900 4200 0900 0.4900 0.9100 1.0000 3600 4B00 1600 0,3600 0.8400 1.0000 2500 5000 2500 0.2500 0.7500 1.0000 _-) 0 1 2 _1 1290 2430 0210 0010 0.1290 0.9720 0.9990 1.0000 5I20 3840 0960 0080 0.5120 0.8960 0.9920 1.0000 3430 44I0 l 890 0210 0.3430 0.7B40 0.9730 1.0000 2160 4320 2880 0640 0.2160 0.64B0 0.9360 1.0000 1250 3150 3750 1250 0.1250 0.5000 0.8750 1.0000 4 0 l 2 J 4 656 1 2916 0486 0036 000l 0.6561 0.947,7 0.9963 0.9999 1.0000 4096 4096 1536 0256 0016 0.4096 0.BI92 0.9,728 0.9984 1.0000 2401 4I16 2646 0156 0081 0.240I 0.6511 0.9l63 0.9919 1.0000 I296 3456 3456 l 536 0256 0.1296 0.4152 0,8208 0.9744 1.0000 0625 2500 3750 2500 0625 0.0625 0.3I25 0.6875 0.9375 1.0000 5 0 l 2 _] 4 5 5905 328l, 0729 0081 0005 0000 0.5905 0.9l85 0.9914 0.9995 1.0000 1.0000 32,17 4096 2048 0512 0064 0003 0.3211 0.1313 0.9421 0.9933 0.9991 1.0000 1681 3602 3087 l323 0284 0024 0.1681 0.5282 0.8369 0.9692 0.9916 1.0000 0118 2592 3456 2304 0768 0102 0.0178 0.3370 0.6826 0.9130 0.9898 1.0000 03l3 l 563 3I25 3125 l 563 03l3 0.03l3 0.1 875 0.5000 0.8l25 0.9688 1.0000 6 0 l 2 _,) 4 5 6 5314 3543 0984 0I46 0012 000l 0000 0.53l4 0.8857 0.9841 0.9987 0.9999 1.0000 1.0000 2621 3932 2458 0819 0154 0015 0001 0.262I 0.6554 0.9011 0.9830 0.9984 0.9999 1,0000 11"76 3025 3241 1 852 0595 0102 0007 0.I176 0.4202 0.1443 0.9295 0.9891 0.9993 1.0000 0461 1 866 3110 2165 l382 0369 0041 0.046] 0.2333 0.5443 0.8208 0.9590 0.9959 1.0000 0156 0938 2344 3l25 2344 0938 0156 0.0l56 0.1 094 0.3438 0.6563 0.8906 0.9844 1.0000 1 0 l 2 J 4 5 6 1 4783 3,720 1240 0230 0026 0002 0000 0000 0.4]83 0.8503 0.9743 0.9973 0.9998 1.0000 1.0000 1.0000 2091 3610 2753 Il47 0287 0043 0004 0000 0.2097 0.5161 0.8520 0.9667 0.9953 0.9996 1.0000 1.0000 0824 241 1 3111 2269 0972 0250 0036 0002 0.0824 0.3294 0.64,71 0.8]40 0.9112 0.9962 0.9998 1.0000 0280 1 306 2613 2903 1 935 0174 0112 00l 6 0.02B0 0.1_586 0.4199 0.7I02 0.9037 0.9BI2 0.9984 1.0000 007B 0547 1641 2134 2134 I64l 054,7 0078 0.0078 0.0625 0.2266 0.5000 0.7734 0.9315 0.9922 1.0000 8 0 l 2 J 4 5 6 7 8 4305 3826 l488 033 1 0046 0004 0000 0000 0000 0.4305 0.8131 0.9619 0.9950 0.9996 1.0000 1.0000 1.0000 1.0000 1678 3355 2936 I468 0459 0092 001 1 0001 0000 0. l678 0.5033 0.7969 0.9437 0.9896 0.9988 0.9999 1.0000 1.0000 0576 l971 2965 254I 136l 0461 0100 0012 0001 0.0516 0.2553 0.5518 0.B059 0.9420 0.9887 0.9987 0.9999 1.0000 0l68 0896 2090 2787 2322 1239 0413 0079 0007 0.0168 0.1064 0.3154 0.594I 0.8263 0.9502 0.99l5 0.9993 1.0000 0039 0313 I094 2188 2734 21 88 I094 0313 0039 0.0039 0.0352 0.1445 0.3633 0.6367 0.8555 0.9648 0.996l 1.0000 APP. 5 Tables A97 Table A6 Poisson Distribution Probability function /(x) fsee (5), Sec. 24.7] and distribution function F(x) I p: 0.| í(x) l ot"l p: 0.2 í(.r)l .r"l p: 0.3 í(x)l or"l l_L : 0.4 í(xli or"l pL : 0.5 í(x) i or_rl 0 1 2 J 4 5 0. 9048 0905 0045 0002 0000 0.9048 0.9953 0.9998 1.0000 1.0000 0. 8l 87 1637 0l64 00l l 0001 0.8l87 0.9825 0.99B9 0.9999 1.0000 0. 1408 2222 0333 0033 0003 0.7408 0.9631 0.9964 0.9991 1.0000 0. 6703 2681 0536 0072 0007 0001 0.6703 0.9384 0.9921 0.9992 0,9999 1.0000 0. 6065 3033 0758 0l26 0016 0002 0.6065 0.9098 0.9856 0.9982 0.9998 1.0000 x p-0.6 /(r) l .r."l LL : 0.1 í(x)l ,trl p:0.8 /(,r) I orrl tL - 0.9 í(x)| .rrl í(x) l F(x) 0 1 2 J 4 5 6 1 0. 5488 3293 0988 0l98 0030 0004 0.54BB 0.87B 1 0.9169 0.9966 0.9996 1.0000 0. 4966 3416 12I1 0284 0050 0007 0001 0.4966 0.8442 0.9659 0.9942 0,9992 0.9999 1.0000 0. 4493 359.5 l 43B 0383 00]1 0012 0002 0,4493 0.808B 0.9526 0.9909 0.9986 0.9998 1.0000 36_59 I647 o494 0l 1l 0020 0003 0. 1066 0.4066 0.1725 0.93,7l 0,9865 0.9911 0.9997 1.0000 0. 3619 3679 l 839 0613 0153 0031 0005 0001 0.3619 0.7358 o.9l91 0.9810 0.9963 0.9994 0.9999 1.0000 X p: 1.5 f(.r) l or"l p:2 .f (x) | .rrl p:3 /(x) I or"l l_L: 4 /(r) l or"l p:5 í(x) l "r"l 0 l 2 J 4 5 6 ] 8 9 10 l1 I2 13 I4 15 16 0. 2231 334] 2510 1255 041 I 0l4l 0035 0008 000l 0.2231 0.5578 0.8088 0.9344 0.9814 0.9955 0.9991 0.9998 1.0000 0. l 353 2107 2101 l804 0902 036l 0l20 0034 0009 0002 0.1353 0.4060 0.6761 0.857l 0.9473 0.9834 0.9955 0.99B9 0.9998 1.0000 0. 0498 1494 2240 2240 1680 l 008 0504 02I6 008 1 0027 0008 0002 0001 0.0498 0. l 99l 0.4232 0.64,72 0.B 153 0.9161 0.9665 0.9881 0.9962 0.9989 0.9991 0.9999 1.0000 0. 0l83 0733 I465 1954 1954 1 563 I042 0595 029B 0132 0053 00l9 0006 0002 000l 0.0183 0.09l6 0.23B l 0.4335 0.6288 0.785l 0.8893 0.9489 0.9786 0.9919 0.9972 0.9991 0.9991 0.9999 1.0000 0. 006,7 0331 0842 1404 1755 1755 I462 I044 0653 0363 0l81 0082 0034 0013 0005 0002 0000 0.0067 0.0404 0.1241 0.2650 0.4405 0.6160 0.]622 0.8666 0.9319 0.9682 0.9863 0.9945 0.9980 0.9993 0.9998 0.9999 1.0000 A98 APP. 5 Tables Table A7 Normal Distribution Values of the distribution function O(z) lsee (3), Sec. 24.8l. O(-z) : 1 - O(z) z O(e) 7 O(z) z ó(z) z O(z) z ó(z) Z O(z) 0,01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0,10 0.11 0.I2 0.13 0.14 0.15 0. l6 0.I1 0.18 0.19 0.20 0.2l 0.22 0.23 0.24 0.25 0.26 0.27 0.28 0.29 0.30 0.31 0.32 0.33 0.34 0.35 0.36 0.37 0.38 0.39 0.40 0.4I 0.42 0.43 0.44 0.45 0.46 0,47 0.48 0.49 0.50 0. 5040 5080 5l20 5160 5I99 5239 5279 5319 5359 5398 5438 5478 5511 5557 5596 5636 5675 5714 5]53 5793 5832 5871 5910 5948 5987 6026 6064 6103 6I4l 6119 6217 6255 6293 633l 6368 6406 6443 6480 65I7 6554 659I 6628 6664 6,700 6136 6772 6808 6844 6879 6915 0.51 0.52 0.53 0.54 0.55 0.56 0.5,7 0.58 0.59 0.60 0.61 0.62 0.63 0.64 0.65 0.66 0.61 0.68 0.69 0.70 0.]I 0.72 0.73 0.,74 0.15 0.76 0.7,7 0.78 0.19 0.80 0.81 0.82 0.83 0.84 0.85 0.86 0.87 0.88 0.89 0.90 0.91 0.92 0.93 0.94 0.95 0.96 0.91 0.98 0.99 1.00 0. 6950 6985 7079 7054 7088 1I23 7I57 1I90 1224 ,7257 729I -l324 7357 1389 7422 7454 7486 15I7 ,7549 7580 161l 7642 7673 7704 1134 7764 1194 ,7823 7852 788 1 79I0 7939 7961 ,7995 8023 805 1 8078 8106 8133 8 159 8186 82l2 8238 8264 8289 83 15 8340 8365 8389 8413 1.01 I.02 1.03 1.04 1.05 1.06 1.01 1.0B 1.09 1.10 1.11 1.12 1.13 1.14 1.15 1.16 1.11 1.18 1.19 1.20 I.2I I.22 I.23 1.24 1.25 |.26 I.21 1.28 I.29 1.30 1.31 1.32 I.33 1.34 1.35 1.36 I.31 1.38 1.39 1.40 I.4l 1.42 I.43 1.44 1.45 1.46 1.47 1.48 I.49 1.50 0. 8438 846I 8485 8508 853 1 8554 85,71 8599 862I 8643 B665 8686 8708 8729 8749 87,70 8190 88 10 8830 8849 8869 8888 8907 8925 8944 8962 8980 8997 9015 9032 9049 9066 9082 9099 9115 913l 9I47 9162 9117 9192 9207 9222 9236 925l 9265 9279 929z 9306 9319 9332 1.51 1.52 1.53 1.54 1.55 1.56 1.57 1.58 1.59 l,60 1.6l I.62 1.63 1.64 1.65 1.66 I.67 1.68 I.69 1.70 1.7I I.72 r.73 1.]4 1.,7 5 I.16 I.,71 1.78 I.19 1.80 1 .81 1.82 1.83 1.84 1.85 1.86 1.87 1.B8 1.89 1.90 1.9l 1.92 1.93 1.94 1.95 1.96 I.97 1.98 1.99 2.00 0. 9345 9357 9370 9382 9394 9406 94I8 94z9 944I 9452 9463 9414 9484 9495 9505 9515 9525 9535 9545 9554 9564 9513 9582 959I 9599 9608 9616 9625 9633 964l 9649 9656 9664 967I 96-18 9686 9693 9699 9106 9713 9719 9726 9732 9738 9144 9150 9756 976I 9761 97,72 2.0l 2.02 2.03 2.04 2.05 2.06 2.01 2.08 z.09 2.10 2.1I 2.12 2.13 2.14 2.15 2.16 2.I7 2.18 2.I9 2.20 2.2I 2.22 2.23 ))4 2.25 2.26 2.27 2.28 2.29 2.30 2.3I 2.32 2.33 2.34 2.35 2.36 2.37 2.38 2.39 Z.40 2.4I .l Aa 2.43 2.44 2.45 2.46 2.47 2.48 2.49 2.50 0. 9778 9783 9788 9793 9798 9803 9808 9812 981,7 982I 9826 9830 9834 9838 9842 9846 9850 9854 9857 9861 9864 9868 981I 9875 9878 988 1 9884 9887 9890 9893 9896 9898 9901 9904 9906 9909 99II 9913 9916 99l8 9920 9922 9925 9921 9929 9931 9932 9934 9936 9938 2.51 2.52 2.53 2.54 2.55 z.56 2.51 2.58 2.59 2.60 2.6L 2,62 2.63 2.64 2.65 2.66 2.61 2.68 2.69 2.70 2.11 2.-12 z.73 2.14 2.75 2.16 2.11 2.78 2.79 2.80 2.8l 2.82 2.83 2.84 2.85 2,86 2.81 2.88 2.89 2.90 2.9I 2.92 2.93 2.94 2.95 2.96 2.9] 2.9B 2.99 3.00 0. 9940 9941 9943 9945 9946 9948 9949 995I 9952 9953 9955 9956 9951 9959 9960 996I 9962 9963 9964 9965 9966 9961 9968 9969 9970 997I 9912 9913 99,74 9974 9975 99]6 9977 9977 9918 99]9 9979 9980 998 1 998 1 9982 9982 9983 9984 9984 9985 9985 9986 9986 9987 APP. 5 Tables Table A8 Normal Distribution Values of e for given values of O(z) [see (3), Sec. 24.8] and D(z) : Q(z) Example: z : 0.279 if a(z) : 617o; z : 0.860 if D(z) : 6I%o. 7o z(O) z@) 7a z(O) z(.D) 7o z(O) z(D) l 2 J 4 5 6 7 8 9 10 l1 I2 13 14 15 16 17 18 19 20 2I 22 z3 24 25 26 2,7 28 29 30 31 32 JJ 34 35 36 37 38 39 40 -2.326 -2.054 - 1.881 - I.7 5I -1.645 - 1.555 - I.416 - 1.405 -I.34l -1.282 - I.227 -1.I75 -1.126 - 1.080 - 1.036 -0.994 -0.954 -0,9l5 -0.878 -0.842 -0.806 -0.112 -0.739 -0.106 -0.674 -0.643 -0.6l3 -0.583 -0.553 -0.524 -0.496 -0.468 -0.440 -0.4I2 -0.385 -0.358 -0.332 -0.305 -0.219 -0.253 0.013 0.025 0,038 0.050 0.063 0.075 0.088 0.100 0.113 0.126 0.1 38 0.151 0.164 0.116 0.1 89 0.202 0.215 0.228 0.240 0.253 0.266 0.2"]9 0.292 0.305 0.319 0.332 0.345 0.358 0.372 0.385 0.399 0.4I2 0.426 0.440 0.454 0.468 0.482 0.496 0.510 0.524 4I 42 43 44 45 46 41 48 49 50 5l 52 53 54 55 56 5,7 58 59 60 61 62 63 64 65 66 61 68 69 70 ,71 12 73 "74 15 16 11 7B 79 B0 -0.228 -0,202 -0.I,76 -0.151 -0.126 -0.100 -0.075 -0.050 -0.025 0.000 0.025 0.050 0.075 0.1 00 0.126 0.15l 0.116 0.202 0.228 0.253 0,2]9 0.305 0.332 0.358 0.385 0.412 0.440 0.468 0.496 0.524 0.553 0.5B3 0.6l3 0.643 0.674 0;706 0.139 0.]12 0.806 0.842 0.539 0.553 0.568 0.583 0.598 0.613 0.628 0.643 0.659 0,674 0.690 0.706 0.722 0.739 0.755 0.772 0.189 0.806 0.824 0.B42 0.860 0.878 0.896 0.915 0.935 0.954 0.914 0.994 1.015 1.036 1.058 1.080 1.103 1.126 1.150 1.175 I.200 1.22,7 I.254 1.2B2 81 82 83 B4 85 86 81 88 B9 90 91 92 93 94 95 96 91 91.5 98 99 0.878 0.915 0.954 0.994 1.036 1.080 1.126 1.115 I.221 1.282 1t341 1.405 1.476 1.555 1.645 1.151 1 .881 1,960 2.054 2.326 1.3l 1 1.341 I.372 1.405 1.440 I.4,76 1.5I4 1.555 1.598 1.645 1.695 L751 1.8l2 1.881 1.960 2.054 2.I,70 2.241 2.326 2.576 99.1 99,2 99.3 99.4 99.5 99.6 99.1 99.8 99.9 2.366 2.409 2.457 2.512 2.516 2.652 2.148 2.B18 3.090 2.612 2.652 2.691 2.748 2.807 2.818 2.968 3.090 3.29I 99.9I 99.92 99.93 99.94 99.95 99.96 99.97 99.98 99.99 3.I21 3,156 3.195 3.239 3.291 3.353 3.432 3.540 3.119 3.320 3.353 3.390 3.432 3.48I 3.540 3.615 3.,719 3.891 Al00 APP. 5 Tables TableA9 f-Distribution Values of z for given values of the distribution function F(z) (see (8) in Sec. 25.3). Example: For 9 degrees of freedom, z: 1.83 when F(z) : 0.95. F(z) Number of Degrees of Freedom : 4',56i1 2 J ,1 8 9 10 0.5 0.6 0.,7 0.8 0.9 0.95 0.915 0.99 0.995 0.999 0.00 0.32 0.13 1.38 3.08 6.31 12.7 31.8 63.1 318.3 0.00 0.29 0.62 1.06 1.89 2.92 4.30 6.96 9.92 22.3 0.00 0.28 0.58 0.98 1.64 2.35 3.1B 4.54 5.84 10.2 0.00 0.2,7 0.51 0.94 1.53 2.13 2.78 3.75 4.60 7.1,7 0.00 0.27 0.56 0.92 1.48 2.o2 2.51 3.36 4.03 5,89 0.00 0.26 0.55 0.9 | 1.44 I.94 2.45 3.14 3.,71 5.2I 0.00 0.26 0.55 0,90 I.4I 1,89 2.36 3.00 3.50 4.79 0.00 0.26 0.55 0,89 I.40 1.86 2.31, 2.90 3.36 4.50 0.00 0.26 0.54 0.88 1,38 1.83 2,26 2.82 3.25 4.30 0.00 0.26 0.54 0.88 1.31 1.81 2.23 2.16 3.11 4.I4 F(.z) Number of Degrees of Freedom 14 i ,, | ,u I l1 l2 l3 I1 18 19 20 0.5 0.6 0.7 0.8 0.9 0.95 0.975 0.99 0.995 0.999 0.00 0.26 0.54 0.88 I.36 1.80 2.20 2.12 3.1 1 4.02 0.00 0,26 0.54 0.B7 1.36 1.78 2.18 2.68 3.05 3.93 0.00 0.26 0.54 0.B7 1.35 1.77 2.16 2.65 3.01 3.85 0.00 0.26 0.54 0.87 1.35 L76 2.14 2.62 2.98 3.19 0.00 0.26 0.54 0.87 I.34 I.75 2.I3 2.60 2.95 3.73 0.00 0.26 0.54 0.86 1.34 1.75 2.12 2.58 2.92 3.69 0.00 0.26 0.53 0.86 1.33 I.14 2.11 2.51 2.90 3.65 0.00 0.26 0,53 0.86 1.33 1.13 2.I0 2.55 2.88 3.61 0.00 0.26 0.53 0.86 I.33 I.73 2.09 2.54 2.86 3.58 0.00 0.26 0.53 0.86 1.33 I.72 2.09 2.53 2.85 3.55 Number of Degrees of Freedom F(z) 0.5 0.6 0.7 0.8 0.9 0.95 0.975 0.99 0.995 0.999 22 0.00 0.26 0.53 0.86 1.32 I.,72 2.01 2.51 2.82 3.50 24 0.00 0.26 0.53 0.86 1.32 1.]1 2.06 2.49 2.80 3.41 26 0.00 0.26 0.53 0.86 1.31 1.1l 2.06 2.48 2.18 3.43 28l:ol+oIso 100 200 @ 0.00|o.oolo.oo 0.00 0.25 0.53 0.B5 1.30 0.00|0.00|0.00 0.26 |o.z0 l o.za 0.25 l0.25 l0.25 0.53|o.s:Io.s: 0.53 l0.53 l0,52 0.85|0.85|o.ss 0.85 l0.B4 l0.84 1.31 l t.:t l t.ro 1.29 l t.zq I l.zs l .70 l 1.70 l l.ó8 1.68 | 1.66 | t.Os | 1.65 2.05 l2.04 lz.oz 2.0l | 1.98 | 1.9] | 1.96 2,47 lz.+0 lz.+z 2.40|2.36|2.35|2.33 2.16 lz.lslz.lo 2.68| 2.63| 2.60 l2.5B 3.4l l :.:q | :.s t 3.26 l 3.17 l 3. l3 l 3.09 APP. 5 Tables Alol Table Al0 Chi-square Distribution Values of x for given values of the distribution function F(7) (see Sec. 25.3 before (17)). Example: For 3 degrees of freedom, z : 11.34 when F(z) : 0.99. In the last column, h : \/2," - 1, where ,, is the number of degrees of freedom. F(z) J Number of Degrees of Freedom l'l,|,i, 1098 1 2 0.005 0.0l 0.025 0.05 0.95 0.975 0.99 0.995 0.00 l o.ot 0.00 l o.ou 0.00 | o.os 0.00 l o. to l 3.84 | ,.nn 5.02 l 7.38 6.63 l 9.21 7.88 l ro.oo 0.01 0.1 1 0.22 0.35 7.8l 9.35 l1.34 12.84 0.2I 0.30 0.48 0.71 9.49 1I.14 13.28 l4.86 0.41 0.55 0.83 1.15 11.0,7 12.83 15.09 16.15 0.6B 0.87 1.24 1.64 12.59 l4.45 16.8l l8.55 0.99 1.24 1.69 2.I,7 14.01 16.01 l8.4B 20.28 1.34 1.65 2.18 2."13 15.51 1].53 20.09 21.95 1.73 l 2.16 2.o9 l z.sa 2.70 l z.zs 3.33 | z.o+ 16.92 l ,r.,, t9.02 | zo.+a 21.61 l zz.zl 23.59 l zr.rg F(z) Number of Degrees of Freedom | 'o|,r|,u|" 20l91813l2l1 0.005 0.01 0.025 0.05 0.95 0.975 0.99 0.995 2.60 l 3.07 3.o5 | :.sz 3.82 | +.+o 4.57 l lzl l l9.68 I z1,or 21.92 l zz.z+ 24.72 l ze .zz 26.76 l zs.:o 3.57 4.1l 5.01 5.89 22.36 24.]4 27.69 29.82 4.01 4.66 5.63 6.51 23.68 26.12 29,14 31.32 4.60 5.23 6.26 ].26 25.00 21.49 30.58 32.80 5.14 5.81 6.91 7.96 26.30 28.85 32.00 34.21 5.10 6.41 7.56 8.67 2].59 30.1 9 33.41 35.12 6.26 1.01 B.23 9.39 28.B1 31.53 34.81 37.16 6.84 l l.+s 7.63 l g.zn 8.91 | o.sl 10.12 l to.ss l 30.14 l zl.+l 32.85 l s+.ll 36.19 l zl.sl 38.58 l +o.oo F(z,) 30292823222l Number of Degrees of Freedom |z^|zslral,0.005 0.01 0.025 0.05 0,95 0.9,I5 0.99 0.995 B.0 l 86 8.9 l q., 10.3 l l1.0 11.6 l n.: I 32.1 l :3.s 35.5 l :o.s 38.9 l +o.: 41.4 l oz.s 9.3 10.2 11.1 l3.1 35.2 38.1 41.6 44.2 9.9 10.9 12.4 13.8 36.4 39.4 43.0 45.6 l0.5 11.5 13. 1 14.6 37.1 40.6 44.3 46.9 11.2 12.2 l3.8 15.4 38.9 41.9 45,6 48.3 11.8 12.9 14.6 16.2 40.1 43.2 4].0 49.6 12.5 13.6 l5.3 16.9 41.3 44.5 48.3 51,0 l3. 1 14.3 16.0 11.] 42.6 45.7 49.6 52.3 13.8 l5.0 l6.8 18.5 43.8 47.0 50,9 53.1 F(z) 605040 Number of Degrees of Freedom zolsolqol100 > l00 (Approximation) 0.005 0.0l 0.025 0.05 0.95 0.975 0.99 0.995 20.7 l zs.o 22.2 l zs.,l 24.4 l zz.+ 26.5 l :+.s I 55.8 l el.s 59.3 | lt.+ 63.7 l 76.2 66.8 l D.s 35.5 37.5 40.5 43.2 79.1 83.3 88.4 92.0 43.3 45.4 48.8 5l,7 90.5 95.0 l00.4 104.2 51.2 53.5 5].2 60.4 101.9 l06.6 1I2.3 116.3 59.2 61.8 65.6 69.1 1 l3.1 l 18.1 124.1 128.3 61.3 l Žtn-2.5812 7o.1 l .].rn-2.3312 74.2 l +rn-1.9612 7].g l iln-1.6412 I 124.3 l Ltn+1.o412 129.6 l +rn+1.96)2 l35.B l iln+2.3312 140.2 l L,n -r 2.58)2 Al02 APP. 5 Tables Table A11 F-Distribution with \m, n| Degrees of Freedom Values of z for which the distribution function F(z) lsee (13), Sec. 25.4] has the value 0.95 Example: For (7, 4) ď.f., z: 6.09 tf F(z) : 0.95. n m:I m:2 m:3 m:4 m:5 m:6 nt:'7 m:8 m:9 1 2 _1 4 5 6 ,7 8 9 10 11 I2 13 l4 15 16 l7 18 I9 20 22 24 26 28 30 32 34 36 38 40 50 60 10 B0 90 100 150 200 1000 a 161 18.5 10.1 7.71 6.6I 5.99 5.59 5.32 5.I2 4.96 4.84 4.75 4.61 4.60 4.54 4.49 4.45 4.4I 4.38 4.35 4.30 4.26 4.z3 4.20 4.17 4.I5 4.I3 4.II 4.I0 4.08 4.03 4.00 3.98 3.96 3.95 3.94 3.90 3.89 3.85 3.84 200 19.0 9.55 6.94 5.19 5.14 4.74 4.46 4.26 4.10 3.98 3.89 3.81 3.14 3.68 3.63 3.59 3.55 3.52 3.49 3.44 3.40 3.3l 3.34 3.32 3.29 3.28 3.26 3.24 3.23 3.18 3.15 3.13 3.1 1 3.10 3.09 3.06 3.04 3.00 3.00 216 19.2 9.28 6.59 5.4I 4.]6 4.35 4.07 3.86 3.11 3.59 3.49 3.4l 3.34 3.29 3.24 3.20 3.16 3.13 3. l0 3.05 3.01 2.98 2.95 2.92 2.90 2.88 2.87 z.85 2.84 2.19 2.]6 2.14 2.72 2.11 2.10 2.66 2.65 2.6I 2.60 225 19.2 9.12 6.39 5.19 4.53 4.I2 3.84 3.63 3.48 3.36 3.26 3.18 3.1 1 3.06 3.0l 2.96 2.93 2.90 2.B,7 2.82 2.78 2.74 2.7I 2.69 2.67 2.65 2.63 2.62 2.6I 2.56 2.53 2.50 2.49 2.41 2.46 2.43 1 Á.) 2.38 2.3,7 230 19.3 9.01 6.26 5.05 4.39 3.97 3.69 3.48 J.JJ 3.20 3.1 l 3.03 2.96 2.90 2.85 2.8I 2.77 2."l4 2.7l 2.66 2.62 2.59 2.56 2.53 2.5I 2.49 2.48 2.46 2.45 2.40 2.37 2.35 2.33 2.32 2.31 2.21 2.26 2.22 2.2r 234 19.3 B.94 6.I6 4.95 4.28 3.B1 3.58 3.37 3.22 3.09 3.00 2.92 2.85 2.79 2.14 2.10 2.66 2.63 2.60 2.55 2.51 2.47 2.45 1A1 2.40 2.38 2.36 2.35 2.34 2.29 2.25 2.23 2.2I 2.20 2.19 2.16 2.I4 2.II 2.I0 237 19.4 8.89 6.09 4.8B 4.21 3.,79 3.50 3.29 3.I4 3.01 2.91 2.83 2.76 2.]1 2.66 2.61 2.58 2.54 2.5I 2.46 2.42 2.39 2.36 2.33 2.3I 2.29 2.28 2.26 2.25 Z.20 2.í1 2.I4 2.13 2.I1 2.10 2.07 z.06 2.02 2.01 239 19.4 8.85 6.04 4.B2 4.I5 3.]3 3.44 J.ZJ 3.07 2.95 2.85 2.17 2.10 2.64 2.59 2.55 2.5I 2.48 2.45 2.40 2.36 2.32 2.29 2.27 2.24 2.23 2.2I 2.19 2.18 2.I3 2.I0 2.07 2.06 2.04 2.03 2.00 1.98 1.95 I.94 24I 19.4 8.81 6.00 4.77 4.10 3.68 3.39 3.1B 3.02 2.90 2.B0 2;71 2.65 2.59 2.54 2.49 2.46 1A1 2.39 2.34 2.30 2.21 2.24 2.21 2.19 2.11 2.15 2.I4 2.I2 2.0,7 2.04 2.02 2.00 I.99 I.97 I.94 I.93 1.89 1.88 APP. 5 Tables Table A11 F-Distribution with Im, r| Degrees of Freedom (continued) Values of z for which the distribution function F(z) [see (13), Sec. 25.4] has the value 0.95 n m: 10 m:15 m:20 m: 30 m: 40 m: 50 m: 10O @ 11 242 2| 9.4 3 l 8.79 4 l _5.96 5 | 4.74 I 6 I 4.06 7 l 3.64 B I 3.3.5 9 l 3.,l4 ,o | 2g8 11 L.r, ,, | 2.75 13 l 2.61 M l 2.60 15 l 2,54 I 16 l 2.4g I7 l 2.45 18 l 2.41 19 l 238 20 l 235 I 22 l 230 24 l 2.25 26 l 2.22 28 l 2.Ig ,o | 2.16 ,rl 214 z+ l 2.12 36 l 2.11 38 l 2.og oo | 2.o8 ,o | 2.o3 60 l 1.gg 70 l 1,g1 80 l 1.95 90 l Lg4 100 150 200 1000 1.93 1.89 1.88 1.84 1.83 246 19.4 B.70 5.86 4.62 3.94 3.51 3.22 3.01 2.85 2.,72 2.62 2.53 2.46 2.40 2.35 2.31 2.27 2.23 2.20 2.15 2.11 2.01 2.04 2.01 1.99 1.9,7 1.95 1.94 1.92 1.87 1.84 1.B 1 I.79 1.78 L71 1.73 I.72 1.68 I.61 248 19,4 8.66 5.80 4.56 3.B1 3.44 3.15 2.94 2.71 2.65 2.54 2.46 2.39 Z.J3 2.28 2.23 2.I9 2,16 2.12 2.01 2.03 1.99 1.96 1.93 1.9t 1.B9 1.87 1.B5 1.84 1.78 1.]5 1.12 1.70 I.69 1.68 1.64 1.62 1.5B 1.5] 250 l9.5 8,62 5.15 4.50 3.B 1 3.3B 3.08 2.86 2.70 2.57 2.41 2.38 2.3I 2.25 2,19 2.I5 2,11 2.01 2.04 1.98 1.94 1.90 1.87 1,84 I.82 1.80 1.78 1.16 I.14 I.69 1.65 1.62 1.60 1.59 1.51 I.54 1.52 1.41 L46 251 19.5 8.59 5.72 4.46 3.]1 3.34 3.04 2.83 2.66 2.53 2.43 2.34 2.27 2.20 2,15 2.10 2.06 2.03 1.99 I.94 1.89 1.85 I.82 I.,79 1.71 I.15 1.13 I.7I I.69 1.63 1.59 1.57 I.54 1.53 1.52 1.48 I.46 1.41 I.39 252 19.5 8.58 5.10 4.44 3.75 3.3 z 3.02 2.B0 2.64 2.51 2.40 2.3I 2.24 2.IB 2.12 2.08 2.04 2.00 I.91 1.9l l.B6 I.82 1.79 1."I6 I.14 I.7I I.69 1.6B 1.66 1.60 1.56 1.53 1.51 1.49 1.48 1.44 1.4I 1.36 1.35 3.71 3.2,I 2.91 2.]6 2.59 2.46 2.35 2.26 2.I9 2.I2 2.07 2.02 1.98 I.94 1.9l 1.85 1.80 1.16 I.13 1.10 1.6,7 1.65 1.62 1.61 1.59 I.52 1.48 I.45 1.43 I.4I I.39 1.34 1.32 L26 1.24 3.61 3.23 2.93 2.1I 2.54 2.40 2.3o 2.2l 2.I3 2.07 2.ol 1.96 1.92 l.BB 1.84 1.78 1.73 I.69 1.65 1.62 1.59 I.57 1.55 1.53 1.51 1.44 I.39 1.35 I.32 1.30 1.28 1.22 I.I9 1.08 1.00 253 l zs+ 19.5 l r q.s B.55 l s.s: 5.66 | s.o: 4.4I l +.zl At04 APP. 5 Tables Table Al1 F-Distribution with (-, n| Degrees of Freedom (continued) Values of z for which the distribution function F(z) [see (13), Sec. 25.4] has the value 0.99 n m:1 m:2 m:3 m:4 m:5 m:6 m:J nt:B m:9 1 2 J 4 5 6 7 8 9 l0 l1 12 l3 I4 15 l6 11 l8 l9 20 22 24 26 28 30 32 34 36 38 40 50 60 70 80 90 100 l50 200 1000 4052 98.5 34.1 21.2 16.3 13.1 12.2 l 1.3 |0,6 l0.0 9.65 9.33 9.01 B.86 8.68 8.53 8.40 8.29 8.18 8.10 7.95 1.82 ,/.,72 7.64 7.56 1.50 7.44 1.40 7.35 1.3I 1.11 7.08 1.0l 6.96 6.93 6.90 6.81 6.76 6.66 6.63 4999 99.0 30.8 l8.0 13.3 10.9 9.55 8.65 8.02 ,7.56 1.2I 6.93 6,10 6.5l 6.36 6.23 6.1 l 6.01 5.93 5.85 5.72 5.61 5.53 5.45 5.39 5.34 5.29 5.25 5.21 5. 1B 5.06 4.9B 4.92 4.88 4.85 4.82 4.15 4.11 4.63 4.6I 5403 99.2 29.5 16.1 12.I 9.1B 8.45 1.59 6.99 6.55 6.22 5.95 5.14 5.56 5.42 5.29 5.18 5.09 5.01 4.94 4.B2 4.12 4.64 4.51 4.51 4.46 4.42 4.38 4.34 4.31 4.20 4.13 4.01 4.04 4,01 3.9B 3.91 3.BB 3.B0 3.18 5625 99.2 28.1 l6.0 II.4 9. 15 7.85 ].01 6.42 5.99 5.6] 5.4l 5.2l 5.04 4.89 4.77 4.61 4.58 4.50 4.43 4.31 4.22 4.14 4.01 4.02 3.91 3.93 3.89 3.86 3.83 3;72 3.65 3.60 3.56 3.54 3.51 3.45 3.4I 3.34 3.32 5164 99.3 28.2 15.5 11.0 B.75 7.46 6.63 6.06 5.64 5.32 5.06 4.86 4,69 4.56 4.44 4.34 4.25 4.I1 4.10 3.99 3.90 3.82 3.15 3.70 3.65 3.61 3.57 3.54 3.51 3.4l 3.34 3.29 3.26 3.23 3.2I 3,14 3.1 l 3,04 3.02 5859 99.3 21.9 15.2 10.1 8.47 1,I9 6,31 _5.80 5.39 5.01 4.B2 4.62 4.46 4.32 4.2o 4.10 4.0l 3.94 3.81 3.,76 3.67 3.59 3.53 3.4,7 3.43 3.39 3.35 3.32 3.29 3.I9 3.12 3.01 3.04 3.01 2.99 2.92 2.89 2.82 2.B0 5928 99.4 2,7.7 l5.0 10.5 8.26 6.99 6.18 5.61 5.2o 4.89 4.64 4.44 4.28 4.14 4.03 3.93 3.84 3.11 3.10 3.59 3.50 3.42 3.36 3.30 3.26 3.22 3.18 3. l5 3.12 3,02 2.95 2.9I 2.87 2.84 2.82 2.76 2.13 2.66 2.64 59B l 99,4 21.5 14.8 l0.3 8. l0 6.84 6.03 5.47 5.06 4.14 4.50 4.30 4.14 4.00 3.89 3.79 3.11 3,63 3.56 3.45 3.36 3.29 3.23 3.11 J.lJ 3.09 3.05 3.02 2.99 2.89 2.82 2.18 2,14 2.12 2.69 2.63 2.60 2.53 2.51 6022 99.4 21.3 14.1 I0.2 1.98 6.72 5.91 5.35 4.94 4.63 4.39 4.I9 4.03 3.89 3,18 3.68 3.60 3.52 3.46 3.35 3.26 3.1B 3.12 3.07 3.02 2.98 2.95 2.92 2.89 2.78 2.72 2.61 2.64 2.61 2.59 2.53 2.50 2,43 2.41 APP. 5 Tables Table AIl F-Distribution with Im, nl Degrees of Freedo m (continued) Values of e ťor which the distribution function F(z) [see (13), Sec. 25,4) has the value 0.99 32 34 36 38 40 50 60 10 80 90 100 l50 200 1000 an l 2 _-) 4 5 6 1 B 9 l0 ll l2 l3 14 l5 l6 17 18 l9 20 22 24 26 28 30 6l51 99.4 26.9 14.2 9.12 7.56 6.31 5.52 4.96 4.56 4.25 4.01 3,82 3.66 3.52 3.41 J._) | J.l j 3.15 3,09 2.98 2.89 2.8l 2.75 2.70 2.65 2.61 2.58 2.55 2.52 )A) 2.35 2.31 2,27 ) )/, 2.22 2.16 2.13 2.06 2.o4 6056 99.4 2,7.2 14.5 l0.1 7.81 6.62 5.8l 5.26 4.B5 4.54 4.30 4.10 3.94 3.80 3.69 3._59 3,51 3.43 3.3,] 3.26 3.1] 3.09 3.03 2.9B 2,93 2.89 2.86 2.83 2,B0 2.10 2.63 2.59 2.55 2.52 2.50 2.44 2.41 2.34 2.32 6209 99.4 26.1 I4.0 9.55 1.40 6.16 5.36 4.8l 4.4l 4. l0 3.86 3.66 3.51 _1.J / 3.26 3.16 3.08 3.00 2.94 2.83 2.74 2.66 2.60 2.55 2.50 2.46 2.43 2.40 2.31 2.27 2.20 2.I5 2.12 2.09 2.07 2.00 1.91 1.90 1.8B 626I 99.5 26.5 13.8 9.38 1.23 5.99 5.20 4.65 4.25 3.94 3.10 3.51 3.35 3.2l 3.10 3.00 2.92 2.84 2.]B 2.61 2.5B 2.50 2.44 2.39 2.34 2.30 2.26 2.23 2.20 2.10 2.03 1.98 1.94 1.92 1.89 l.83 1.,79 1.12 I.10 6281 99,5 26.4 13.1 9.29 1.14 5.91 5.12 4.51 4.11 3.B6 3.62 3.43 J.l l 3.13 3.02 2.92 2.B4 2.16 2.69 2.58 2.49 1 la /-.+Z 2.35 2.30 2.25 2.21 2,18 2.14 2.11 2.0I 1.94 1.89 1.85 1.82 1.80 1.13 1.69 1.6l 1.59 6303 99.5 26.4 13.1 9.24 7.09 5.86 5.01 4.52 4.12 3.81 3.5,7 3.38 3.22 3.0B 2.91 2.87 2.7B 2,] 1 2.64 2.53 2.44 Z.Jo 2.30 2.25 2.20 2.16 2.12 2.09 2.06 1.95 1.88 1.B3 1.,79 I.76 l.]4 1.66 1.63 I.54 1.52 6334 99.5 26.2 13.6 9.13 6,99 5.15 4.96 4.42 4.01 3.] 1 3.47 3.21 3.1 l 2.98 2.86 2.16 2.68 2,60 2.54 2.42 2.25 2.19 2,13 2.0B 2.04 2.00 1.97 1.94 1.82 1.15 1,10 1.65 1.62 1.60 1.52 1.48 1.3B 1.36 6366 99.5 26.1 l3.5 9.02 6.8B 5.65 4.86 4.3l 3.9I 3.60 3.36 3.I7 3.00 2.B1 2.15 2.65 2.51 2.49 2.42 2.3l 2.21 2.13 2.06 2.0I 1.96 1.91 1.87 l.84 1.80 1.68 1.60 l Al05 A106 APP.5 Tables Table Al2 Distribution Function F(x) = P(r = x) of the Random Variable T in Section 25.8 x n :4 0 1 2 0. 042 I67 375 x n :5 0 1 2 J 4 0. 008 042 II7 242 408 x n :6 0 1 2 _,) 4 5 6 7 0. 001 008 028 068 136 235 360 500 X n - 1 2 J 4 5 6 7 8 9 10 0. 001 005 015 035 068 119 1,9l 28I 386 500 x n -8 2 J 4 5 6 1 8 9 10 11 l2 13 0. 001 003 007 016 031 054 089 138 I99 2,74 360 452 X n :9 4 5 6 7 8 9 10 11 I2 13 I4 15 1,6 l7 0. 001 003 006 012 02z 038 060 090 130 I79 238 306 381 460 x n :10 6 1 8 9 10 11 I2 13 I4 15 16 I7 l8 I9 20 2l 22 0. 001 00z 005 008 0l4 023 036 054 078 108 146 190 242 300 364 43I 500 x lL :11 8 9 10 11 I2 13 I4 15 16 1] 18 I9 20 2I 22 /-3 24 25 26 21 0. 001 002 003 005 008 013 020 030 043 060 082 109 14I l79 223 2,7I 324 381 440 500 X n :20 50 51 52 53 54 55 56 57 58 59 60 6I 62 63 64 65 66 6,7 68 69 10 ,71 ,72 73 74 ,75 ,I6 ,77 78 -l9 80 81 82 83 84 85 86 81 88 89 90 9I 92 93 94 0. 001 002 002 003 004 005 006 007 008 010 012 0I4 0I7 020 023 027 032 031 043 049 056 064 013 082 093 I04 II7 130 I44 159 776 193 211 230 250 2,7I 293 315 339 362 387 411 436 462 481 x n :19 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 6I 62 63 64 65 66 61 68 69 10 7I 72 73 74 75 ]6 77 78 79 80 81 82 83 84 85 0. 001 002 00z 003 003 004 005 006 008 010 012 0I4 0I7 021 025 029 034 040 047 054 062 07z 08z 093 105 I19 I33 I49 I66 I84 203 223 245 267 290 314 339 365 39I 418 445 413 500 x n :18 38 39 40 4I 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 51 58 59 60 61 62 63 64 65 66 67 68 69 10 7I 72 74 75 16 0. 001 002 003 003 004 005 007 009 011 013 016 020 024 029 034 04I 048 056 066 076 088 100 115 130 I47 I65 184 205 221 250 215 300 327 354 383 4II 44l 470 500 x n :I1 3z jJ 34 35 36 37 38 39 40 4I 42 43 44 45 46 41 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 6,| 0. 001 002 002 003 004 005 007 009 011 0I4 0I7 02I 026 03z 038 046 054 064 076 088 l02 118 135 154 í74 196 220 245 2,7I 299 328 358 388 420 452 484 X n :16 27 28 29 30 3I 32 JJ 34 35 36 5l 38 39 40 4I 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 5,,l 58 59 0. 001 002 002 003 004 006 008 010 013 016 02I 026 032 039 048 058 070 083 09,7 I14 I33 153 I75 I99 225 z53 282 313 345 378 412 447 482 X n :15 23 24 25 26 2,7 28 29 30 3I 5/. JJ 34 35 36 31 38 39 40 4I 42 43 44 45 46 47 48 49 50 51 52 0. 001 002 003 004 006 008 010 01,4 018 023 029 03,I 046 057 070 084 101 I20 I4I l64 190 zI8 248 279 313 349 385 4z3 46I 500 x n :14 18 19 20 zI 22 23 24 25 26 27 28 29 30 31 32 JJ 34 35 36 -7 l 38 39 40 4I 42 43 44 45 0. 001 002 002 003 005 007 010 013 018 024 031 040 051 063 0,79 096 1,n 140 165 194 2z5 z59 295 334 314 415 457 500 X n :13 I4 15 I6 I7 18 I9 20 2I 22 23 24 25 26 z7 28 29 30 3I 32 JJ 34 35 36 3,| 38 0. 001 001 002 003 005 007 011 015 02I 029 038 050 064 082 I02 I26 153 l84 2I8 255 295 338 383 429 476 X n :12 11 I2 13 I4 15 I6 I7 18 l9 20 2I 22 z3 z4 25 26 27 28 29 30 31 32 0. 001 002 003 004 007 010 016 022 031 043 058 016 098 I25 155 190 230 273 3|9 369 420 413 PHOTO CREDlTS Part A opener: John Sohm/Chromoshohm/Photo Researchers. Part B Opener: Lester Lefkowitz/Corbis Images. Part C Opener: Science Photo Library/Photo Researchers. Paít D Opener: Lester Lefkowitz/Corbis Images. Part E Opener: Walter Hodges/Stone/Getty Images. Chapter 19, Figure 434: From Dennis Sharp, ARCHITECKTUR. @1973 der deutschsprachigen Ausgabe Edition Praeger GmbH, Munchen. Part F Opener: Charles O'Rear/Corbis Images. Part G Opener: Greg Pease/Stone/Getty Images. Appendix 1 Opener: Patrick Bennett/Corbis Images. Appendix 2 Opener: Richard T. Nowitz/Corbis Images. Appendix 3 Opener: Chris Kapolka/Stone/Getty Images. Appendix 4 Opener: John Olson/Corbis Images. Pl lNDEX Page numbers Al, A2, A3, . refer to App. 1 to App. 5 at the end of the book. A Abel 78 Absolute convergence 66], 697 frequency 994, 1000 value 607 Absolutely integrable 508 Acceleration 395,995 Acceptance samplin g 1073 Adams-Bashforth methods 899 Adams-Moulton methods 900 Adaptive 824 Addition of complex numbers 603 matrices 275 means 1038 normal random variables 1050 power series l74,680 variances 1039 vectors 2]6,324,367 Addition rule 1002 Adiabatic 561 ADI method 915 Adjacency matrix 956 Adjacent vertices 955 Aitfoil732 Airy's equation 552, 904 Algebraic multiplicity 337, 865 Algorithm ]]7,783 Dijkstra 964 efficíent 962 Ford-Fulkerson 979 Gauss 837 Gauss-Seidel 848 Greedy 967 Kruskal 967 Moore 960 polynomially bound ed 962 Prim 97I Runge-Kutta 892, 904 stable, unstable 783 Aliasing 526 Allowable number of defectives 1073 Alternating direction implicit method 915 path 983 Alternative hypothesis 1058 Ampěre 92 Amplification 89 Amplitude spectrum 506 Analytic function I75, 677, 68I Analytic at infinity 711 Angle between curves 35 between vectors 372 Angular speed 38I,765 Annulus 613 Anticommutative 379 AoQ, AOQL 1075_1076 Approximate solution of differential equations 9, 886-934 eigenvalue problems 863-882 equations 787-796 systems of equations 833-858 Approximation least squares 860 polynomiaL 791 trigonometric 502 A priori estimate 794 AQL 1074 Arc of a curve 391 Archimedian principle 68 Arc length 393 Arctan 634 Area 435,442,454 Argand diagram 605 Argument 607 Artificial v ariable 9 49 Assignment problem 982 Associated Legendre functions 182 Asymptotically equal 191, 1009 normal 1057 stable 148 lndex Attractive 148 Augmented matrix 288, 833 Augmenting path 9] 5, 983 theorem 911,984 Autonomous 31, 151 Average (see Mean value) Average outgoing quality 1075 Axioms of probability 1001 B Back substitution 289, 834 Backward differences 807 edge 974,976 Euler method 896,907 Band matrtx 9I4 Bashforth method 899 Basic feasible solution 942, 944 variables 945 Basis 49, 106, 113, 138, 300,325,360 Beam I20,547 Beats 87 Bellman optimality principle 963 Bell-shaped curve 1026 Bernoulli 30 distribution 1020 equation 30 law of large numbers 1032 numbers 690 Bessel 189 equation I89,204 functions 191, 198, 202,207, A94 functions, tables A94 inequality 2l5,504 Beta function 464 Bezier curves 8l6 BFS 960 Bijective mapping 729 Binary 782 Binomial coefficients 1009 distribution 1020, A96 series 689 theorem 1010 Binormal (Fig.2I0) 397 Bipartite graph 982, 985 Birthday problem 1010 Bisection method 796 Bolzano-Weierstrass theorem A9 1 Bonnet 181 Boundary 433 conditions 203,540,558, 571, 587 point 433,613 value problem 203, 558 Bounded domain 646 function 38 region 433 sequence A69 Boxplot 995 Branch cut 632 potnt 746 Breadth first search 960 Buoyance force 68 c Cable 52, I98,593 CAD (Computer aided design) 810 Cancellationlaw 32I Cantor-Dedekind axiom '{69 Capacitance 92 Capacitor 92 Capacity of a cut set9]6 of an edge 973 Cardano 602 Cardioíd 443 Cartesian coordinates 366, 604 CAS (Computer algebra system) vli,777 Catenary 399 Cauchy 69 convergence principle 667, A90 determinant II2 -Goursat theorem 64J -Hadamard formula 6]6 inequality 660 integral formula 654 integral theorem 647, 652 method of steepest descent 938 principal value ] 19, 722 product 680 -Riemann equations 3], 618, 62I -Schwarz inequality 326, 859 Cayley transform atlon J 39 lndex Center I43 of a gtaph 973 of gravity 436, 45] of a power series 171 Central differences 808 limit theorem 1057 moments 1019 Centrifugal, centripetal 396 Cgs system: Front cover Chain rule 401 characteristic determinant 336,864 equation 59, II7, 336,551, 864 function 542,5]4 of a partial differential equation 551 polynomial336, 864 value 324,864 vector 324,864 Chebyshev polynomials 209 Chinese postman problem 963 Chi-square 1055, I0'7], A101 Cholesky's method 843 Chopping 782 Chromatic number 987 Circle 391 of convergence 675 Circuit 95 Circular disk 613 helix 39I,394 membrane 580 Circulation 764 Cissoid 399 Clairaut equation 34 Class intervals 994 Closed disk 613 integration formula 822, 827 interval A69 path 959 point set 613 region 433 Coefficient matrix 288, 833 coefficients of a differential equation 46 power series 171 system of equations 287, 833 Cofactor 309 Collatz's theorem 870 Cohlmn 275 space 300 sum norm 849 vector 2J5 Combination 1007 Combinatorial optim ization 9 5 4-986 Comparison test 668 Complement 613, 988 Complementary error function A64 Fresnel integrals A65 sine integral ,{66 Complementation rule 1002 Complete bipartite graph 987 graph 958 matching 983 orthonormal set 2I4 Complex conjugate numbers 605 exponential function 57, 623 Fourier integral 519 Fourier series 497 function 614 hyperbolic functions 628, 7 43 impedance 98 indefinite integr aI 637 integration 631 -663, ] 12-725 line integral633 logarithm 630, 688 matrices 356 number 602 plane 605 plane, extended 710 potentíal76I sequence 664 series 666 sphere 710 trigonometric functions 626, 688 trigonometric polyn omíal 524 varíable 614 vector space 324,359 Complexity 96I Component 366,3]4 Composite transformation 28 1 Compound interest 9 Compressible fluid 412 Computer aided design (CAD) 810 algebra system (CAS) vii,7]] l3 lndex Computer (Cont.) Convergence (Cont.) graphics 287 of a sequence 386,664 software (see Software) of a series I7I,666 Conchoid 399 superlinear 795 Condition number 855 tests 667-672 Conditionally convergent 667 uniform 69l Conditional probability 1003 Conversion of an ODE to a system 134 Conduction of heat 465, 552,75] Convolution 248, 523 Cone 406,448 Cooling 14 CONF 1049 Coordinate transformations A71. A84 confidence coordinates intervals 1049-1058 Cartesian 366,604 level 1049 curvilinear A]I limits 1049 cylindrical 587, A]1 Conformal mapping 730, ]54 polar I3J, 443,580, 607 Conic sections 355 spherical 588, A71 Conjugate Coriolis acceleration 396 complex numbers 605 Corrector 890, 900 harmonic function 622 Correlation analysis 1089 Connected Cosecant 62], A62 graph 960 Cosine set 613 of a complex variable 621,688,743 Conservative 4I5,428 hyperbolic 688 Consistent equations 292,303 integral A66, A95 Constraints 93] of a real variable ,{60 Consumer's risk 1075 Cotangent 62], A62 Continuity Coulomb 92 of a complex function 615 law 409 equation 413 Covariance 1039, 1085 of a vector function 387 Cramer's rule 306-30],312 Continuous Crank-Nicolson method 924 distribution 1011 Critical random variable 1011, 1034 damping 65 Contour integral 647 point 3I, I42, ]30 Contraction 1B9 region 1060 Control Cross product 377 chart 1068 Crout's method 841 limit 1068 Cubic spline 81l variables 936 Cumulative Convergence distribution function 1011 absolute 667 frequency 994 circle of 675 Curl 414,430,472, A7l conditional661 Curvature 397 interval I]2,676 Curve 389 of an iterative process ]93,848 arc length of 393 mean 2l4 titting 859 mean-square 2I4 orientation of 390 in norm 2I4 piecewise smooth 421 principle 667 rectifiable 393 raďlus 172 simple 391 lndex Curve (Cont.) smooth 421, 638 twisted 391 Curvilinear coordinates A71 Cut set 976 Cycle 959 Cycloid 399 Cyltnder 446 flow around]63,]67 Cylindrical coordinates 587, 'A71 D D'Alembert's solution 549, 551 Damping 64, 88 Dantzig 944 DATA DESK 991 Decay 5 Decreasing sequence A69 Decrement 69 Dedekind A69 Defect 33] Defective item 1073 Definite complex integral 639 Definiteness 356 Deformation of path 649 Degenerate feasible solution 947 Degree of precision 822 a vertex 955 Degrees of freedom 1052, 1055, 1066 Deleted neighborhood ] 12 Delta Dirac 242 Kronecker 210, A83 De Moivre 6l0 formula 610 limit theorem 1031 De Morgan's laws 999 Density 10l4 Dependent linearly 49, J4, 106,29],300,325 random variables 1036 Depth first search 960 Derivative of a complex function 6l6, 658 directional 404 left-hand 484 right-hand 484 of a vector function 387 DERIVE 778 Descartes 366 Determinant 306. 308 Cauchy Il2 characteristic 336, 864 of a matrix 305 of a matrix product 321 Vandermonďe Il2 DFS 960 Diagonalization 351 Diagonal matrix 284 Diagonally dominant matrix 868 Diameter of a graph 973 Differences 802, 804, 807-808 Difference table 803 Differentiable complex function 616 DifferentiaI 19,429 form 20, 429 geometry 3B9 operator 59 Differential equation (ODE and PDE) Atry 552,904 Bernoulli 30 Bessel I89,204 Cauchy-Riemann 3], 618, 62l with constant coefficients 53, 111 elliptic 551, 909 Euler-Cauchy 69, 185 exact 20 homogeneous 27,46, 105, 535 hyperbolic 551,92B hypergeometric l88 Laguerre 257 Laplace 407, 465, 536, 5'79,587, 910 Legendre I]],204, 590 linear 26,45, 105, 535 nonhomogeneous 2J, 46, 105, 535 nonlinear 45, 151, 535 numeric methods for 886-934 ordinary 4 parabolic 551,909,922 partial 535 Poisson 910, 918 separable 12 Sturm-Liouville 203 of vibrating beam 54], 552 of vibrating mass 61, 86, I35, I50,2l 342,499 of vibrating membrane 569-586 of vibrating string 538 243,26I, l5 lndex Differentiation analytic functions 691 complex functions 616 Laplace transforms 254 numeric 827 power series I74,680 series 696 vector functions 387 Diffusion equation 464, 552 Diffusivity 552 Digraph 955 Dijkstra's algorithm 964 Dimension of vector space 300,325,369 Diocles 399 Dirac's delta 242 Directed graph 955 line segment 364 path 982 Directional derivative 404 Direction field 10 Direct method 845 Diríchlet 46] discontinuous factor 509 problem 467, 558, 587,9I5 Discharge of a source 767 Discrete Fourier transform 525 random variable 1011, 1033 spectrum 507, 524 Disioint events 998 Disk 613 Dissipative 429 Distribution 1010 Bernoulli 1020 binomial 1020, A96 chi-square 1055, A101 continuous 1011 discrete 1011, 1033 Fisher's F- 1066, AI02 -free test 1080 function l0II, 1032 Gauss 1026 hypergeometric 1024 marginal 1035 multinomial 1025 normal 1026, 1041-1051, 1062-1067, A98 Poisson 1022, 1013, A91 Student's t- 1053, A100 two-dimensional 1032 Distribution (Cont.) uniform 1015, I0I7, 1034 Divergence theorem of Gauss 459 of vector fields 4I0, A]2 Divergent sequence 665 series I1I,667 Divided differences 802 Division of complex numbers 604, 609 Domain 40I,613,646 Doolittle's method 841 Dot product 325,3]I Double Fourier series 576 íntegral433 labeling 968 precision 782 Driving force 84 Drumhead 569 Duffing equation 159 Duhamel's formula 597 E Eccentricity of a vertex 9J3 Echelon form 294 Edge 955 coloring 987 incidence list 957 Efficient algorithm 9 62 Eigenbasis 349 Eigenfunction 204, 542, 559, 574 expansion 210 Eigenspace 336, 865 Eigenvalue problems for matrices (Chap. 8) 333-363 matrices, numerics 863-882 ODEs (Sturm-Liouville problems) 203-216 PDEs 540-593 systems of ODEs 130-165 Eigenvector 334,864 EISPACK 778 Elastic membrane 340 Electrical network (see Networks) Electric circuit (see Circuit) Electromechanical analogies 96 Electromotive force 91 Electrostatic field 750 lndex Electrostatic (Cont.) potential 588-592,150 Element of matrtx 2J3 Elementary matrix 296 operations 292 Elimination of first derivative 197 Ellipse 390 Ellipsoid 449 Elliptic cylinder 448 paraboloid 448 partial differential equation 551, 909 Empty set 998 Engineering system: Front cover Entire function 624, 66I, 7 lí Entry 273,309 Equality of complex numbers 602 matrices 275 vectors 365 Equally likely 1000 Equipotential Iines 750,762 surfaces 750 Equivalen ce relatíon 296 Equivalent linear systems 292 Erf 568, 690, A64, A95 Error 783 bound 784 estimation 785 function 568, 690, A64, A95 propagation 184 Type I, Type II 1060 Essential singularity 708 Estimation of parameters 1046-1057 Euclidean norm 327 space 327 Euler 69 backward methods for ODEs 896,907 beta function '464 -Cauchy equation 69, 108, 116, 185, 589 -Cauchy method 887, 890, 903 constant 200 formula 58, 496, 624, 627,68] formulas for Fourier coefficients 480,487 graph 963 numbers 690 method for systems 903 Euler (Cont.) trail963 Evaporation 18 Even function 490 Event 99'I Everett interpolation formula 809 Exact differential equation 20 differential form 20, 429 Existence theorem differential equations 37,73, I07, I09, I75 Fourier integral 508 Fourier series 484 Laplace transforms 226 Linear equations 302 Expectation l0l6 Experiment 997 Explicit solution 4 Exponential decay 5 function, complex 57, 623 function, real ,{60 growth 5,31 integral A66 Exposed vertex 983 Extended complex plane 7I0,736 Extension, periodtc 494 Extrapolation ]97 Extremum 937 F Factorial function l92, 1008, A95 Failure 1021 Fair die 1000 Falling body 8 False position 796 Family of curves 5, 35 Faraday 92 Fast Fourier transform 526 F-distribution 1066, A702 Feasible solution 942 Fehlberg 894 Fibonacci numbers 683 Field conservative 4l5,428 of force 385 gravitational 385, 407, 4lI, 587 17 I37, lndex Field (Cont.) Fourier (Cont.) irrotational 4I5, ]65 series, generalized 2I0 scalar 384 sine integral 511 vector 384 sine series 49I,543 velocity 385 sine transform 514, 530 Finite complex plane 710 transform 579,53I,565 First transform, discrete 525 fundamental form 457 transform, fast 526 Green's formula 466 Fractional linear transformatton'l34 shifting theorem 224 Fraction defective 1073 Fisher, R. A. 1047, 1066, 1011 Fredholm 201 F-distribution 1066, AI02 Free Fixed fall 18 decimal point 781 oscillations 61 point 736,78I, ]8] Frenet formulas 400 Flat spring 68 Frequency 63 Floating point 781 of values in samples 994 Flow augmentingpath975 Fresnel integrals 690, A65 Flows in networks 973*981 Friction 18-19 Fluid flow 4I2, 463,76I Frobenius 182 FIux 412,450 method 182 integral 450 norm 849 Folium of Descartes 399 theorem 869 Forced oscillations 84,499 Fulkerson 979 Ford-Fulkerson algorithm 979 Full-wave recttfter 248 Forest 970 Function Form analytic 115,617 Hermitian 361 Bessel 191, 198, 202,207, A94 quadratic 353 beta A64 skew-Hermitian 361 bounded 38 Forward characteristtc 542,5]4 differences 804 complex 614 eďge 974,976 conjugate harmonic 622 Four-color theorem 987 entire 624, 66I,7II Fotrier 4]] error 568, 690, A64, A95 -Bessel series 213,583 even 490 coefficients 480, 487 exponential 5J, 623, A60 coefficients, complex 491 factorial 192, 1008, A95 constants 210 gamma I92, A95 cosine integral 511 Hankel2)2 cosine series 491 harmonic 465, 622, ]72 cosine transform 5l4,529 holomorphic 617 double series 576 hyperbolic 628,743, A62 half-range expansions 494 hypergeometric 188 integral 508, 563 inverse hyperbolic 634 integral, complex 519 inverse trigonometric 634 -Legendre series 2I2,590 Legendre I77 matrtx 525 logarithmic 630, '\60 series 2II, 480, 487 meromorphic 1II series, complex 497 Neumann 201 lndex Function (Cont.) odd 490 orthogonal 205, 482 orthonormal205. 2I0 periodic 478 probability l0I2, 1033 rational6I'7 scalar 384 space 382 staircase 248 step 234 trigonometríc 626, 688, A60 unit step 234 vector 384 Function space 327 Fundamental form 457 matrix 139 mode 542 period 485 system 49, 106, 113, 138 theorem of algebra 662 G Galilei 15 Gamma function 192, A95 GAMS 778 Gauss 188 distribution 1026 divergence theorem 459 elimination method 289, 834 hypergeometric equation 188 integration formula 826 -Jordan elimination 3l], 844 least squares 860, 1084 quadrature 826 -Seidel iteration 846, 9I3 General powers 632 solution 64, 106, 138, 159 Generalized Fourier series 210 function 242 solution 54_5 triangle inequality 608 Generating function I8I, 216, 258 Geometric multiplicity 337 series 16], 668, 673, 687, 692 19 Gerschgorin's theorem 866 Gibbs phenomenon 490, 510 Global error 887 Golden Rule 15, 23 Goodness of fit 1076 Gosset 1066 Goursat 648, A88 Gradient 403, 4I5, 426, A72 method 938 Graph 955 biparttte 982 complete 958 Euler 963 planar 987 sparse 957 weighted 959 Gravitation 385, 407,4I1, 587 Greedy algorithm 967 Greek alphabet: Back cover Green 439 formulas 466 theorem 439,466 Gregory-Newton formulas 805 Growth restrictíon 225 Guldin's theorem 458 H Hadamard's formula 676 Half-life time 9 Half-plane 613 Half-range Fourier series 494 Half-wave rectifier 248. 489 Halving 8l9,824 Hamiltonian cycle 960 Hanging cable 198 Hankel functions 202 Hard spring 159 Harmonic conjugate 622 function 465, 622,7]2 oscillation 63 series 670 Heat equation 464, 536, 553, ]5],923 potential 758 Heaviside 221 expansions 245 formulas 247 !l0 lndex Heaviside (Cont.) functíon 234 Helicoid 449 Helix 39I,394,399 Helmholtz equation 572 Henry 92 Hermite interpolation 816 polynomials 216 Hermitian 357,36I Hertz 63 Hesse's normal form 375 Heun's method 890 High-frequency line equations 594 Hilbert 20I,326 matrix 858 space 326 Histogram 994 Holomorphíc 611 Homogeneous differential equation 2J, 46, 105, 535 system of equations 288, 304,833 Hooke's law 62 Householder' s tridia gonalization 875 Hyperbolic differential equation 55I, 909, 928 functions, complex 628, 1 43 functions, real A62 paraboloid 448 partial differential equations 551, 928 spiral 399 Hypergeometric differential equation 1 88 distribution 1024 functions 188 series 188 Hypocycloid 399 Hypothesis 1058 ! Idempotent matrix 286 Identity of Lagrange 383 matrix (see Unit matrix) theorem for power series 679 transformation ]36 trick 351 Ill-conditioned 851 Image 327, ]29 Imaginary axis 604 part 602 antt 602 Impedance 94,98 Implicit solution 20 Improper integral 222, ]19, ]22 Imp se 242 IMSL 778 Incidence list 957 matrix 958 Inclusion theorem 868 Incomplete gamma function A64 Incompressible 765 Inconsistent equations 292, 303 Increasing sequence A69 Indefinite integral 631, 650 integration 640 Independence of path 426, 648 Independent events 1004 random variables 1036 Indicial equation 184 Indirect method 845 Inductance 92 Inductor 92 Inequality Bessel 2I5,504 Cauchy 660 ML- 644 Schur 869 triangle 326,372, 608 Infinite dimensionaI325 population 1025, 1045 sequence 664 series (see Series) Infinity 7I0,736 Initial condition 6, 48, I37, 540 value problem 6,38, 48, I07,886, 902 Injective mapping 729 Inner product 325, 359, 37 I space 326 Input 26,84,230 Instability (see Stability) Integral contour 647 lndex Integral (Cont.) definite 639 double 433 eqlation 252 Fourier 508, 563 improper 119,722 indefinite 637, 650 ltne 42I,633 surface 449 theorems, complex 647, 654 theorems, real439, 453, 469 transform 22I.5I3 triple 458 Integrating factor 23 Integration complex functions 631 -663, ] 01-7 27 Laplace transforms255 numeric 817-827 power series 680 series 695 Integro-differential equation 92 Interest 9,33 Interlacing of zeros 797 Intermediate value theorem 796 Interpolation 797-8l5 Hermite 816 Lagrange 798 Newton 802, 805, 807 spline 811 Interquartile range 995 Intersection of events 998 Interval closed ,4.69 of convergence I72,676 estimate 1046 open 4, ,{69 Invariant subspace 865 Inverse hyperbolic functions 634 mapping principle 733 of a matrix 3I5, 844 trigonometric functions 634 Inversion 735 Investment 9, 33 Irreducible 869 Irregular boundary 919 Irrotational 4I5, 765 Isocline 10 Isolated singularity 707 Isotherms 758 Iteration for eigenvalues 872 for equations ]87-794 Gauss-Seidel 846, 913 Jacobi 850 Picard 41 J Jacobian 436,733 Jacobi iteration 850 Jerusalem, Shrine of the Book 814 Jordan 316 Joukowski aírfoll732 K Kirchhoff's laws 92, 973 Kronecker delta 210, A83 Kruskal' s algorithm 967 Kutta 892 L ly 12, /- 853 L2 863 Labeling 968 Lagrange 50 identity of 383 interpolation 798 Laguerre polynomials 209, 25] Lambert's |aw 43 LAPACK 778 Laplace 22l equation 40J, 465, 536, 5'79,587, 910 integrals 512 limit theorem 1031 operator 408 transform 22I,594 Laplacian 443, A73 Latent root 324 Laurent series 70I,712 Law of absorption 43 cooling 14 gravitation 385 large numbers 1032 mass actlon 43 the mean (see Mean value theorem) LC-circuit 97 Ill ll2 lndex LCL 1068 Least squares 860, 1084 Lebesgue 863 Left-hand derivative 484 limit 484 Left-handed 378 Legendre l77 differential equation I]], 204, 590 functions 177 polynomials l79, 20], 590, 826 Leibniz 14 convergence test A70 Length of a curve 393 of a vector 365 Leonardo of Pisa 638 Leonttef 344 Leslie model 34l Libby 13 Liebmann's method 913 Likelihood function 1047 Limit of a complex function 615 cycle I57 left-hand 484 point A90 right-hand 484 of a sequence 664 vector 386 of a vector function 387 Lineal element 9 Linear algebra 27I-363 combination 106,325 dependence 49, J4, 106, I08,29],325 differential equation 26, 45, 105, 535 element 394, A72 fractional transform ation ] 34 independence 49, J4, 106, 108,29],325 interpolation 798 operator 60 optimization 939 programming 939 space (see Vector space) system of equations 287, 833 transformation 28I, 327 Linearization of systems of ODEs 151 Line integral42l, 633 Lines of force 751 LINPACK 779 Liouville 203 theorem 661 Lipschitz condition 40 List 957 Ljapunov 148 Local error 887 minimum 937 Logarithm 630, 688, A60 Logarithmic decrement 69 integral A66 spiral 399 Logistic population law 30 Longest path 959 Loss of significant digits 785 Lotka-Volterra population model 154 Lot tolerance per cent defective 1075 Lower control limit l068 triangular matrix 283 LTPD 1075 LU-factorization 84I M Maclaurin 683 series 683 trisectrix 399 Magnitude of a vector (see Length) Main diagonal 274,309 Malthus's law 5, 31 li4.APLE 119 Mapping 32], ]29 Marconi 63 Marginal distributions 1035 Markov process 285,34l Mass-spring system 61, 86, l35, 150, 243,252, 261,342, 499 Matching 985 MATHCAD ]19 MATHEMATICA ]19 Mathematical expectation 1019, 1038 MATLAB 779 Matrix additton 2]5 augmented 288, 833 band 9I4 lndex Matrix (Cont.) diagonal 284 eigenvalue problem 333-363, 863-882 Hermitian 357 identity (see Unit matrix) inverse 315 inversion 315.844 multiplication 2] 8, 32I nonsingular 315 norm 849,854 normal 362,869 null (see Zero matríx) orthogonal 345 polynomial 865 scalar 284 singular 3 15 skew-Hermitian 357 skew-symmetric 283, 345 sparse 8I2,9l2 square 274 stochastic 285 symmetric 283,345 transpose 282 triangular 283 tridiagonal 8l2, 8]5, 914 unit 284 unitary 357 zero 2J6 Max-flow min-cut theorem 978 Maximum 937 flow 979 likelihood method 1047 matching 983 modulus theorem JJ2 principle 773 Mean convergence 2I4 Mean-square convergence 21 4 Mean value of a (an) analytic function 771 distribution 10l6 function ]64 harmonic function ]]2 sample 996 Mean value theorem 402,434,454 Median 994, 1081 Membrane 569-586 Meromorphic function 71 1 Mesh incidence matrix 2]8 Method of false position ]96 It3 Method of (Cont.) least squares 860, 1084 moments 1046 steepest descent 938 undetermined coefficients 78, ll7, 160 variation of parameters 98, 118, 160 Middle qlartile 994 Minimum 937,942, 946 MINITAB 991 Minor 309 Mixed boundary value problem 55B, 587, 759,9I7 triple product 381 Mixing problem 13, l30, 146, 163,259 Mks system: Front cover ML-ínequality 644 Móbius 453 strtp 453,456 transformation 734 Mode 542,582 Modeling 2, 6, 13, 61,84, l30, I59,222,340, 499, 538,569, 750_167 Modified Bessel functions 203 Modulus 607 Molecule 912 Moivre's formula 610 Moment central 1019 of a distribution 1019 of a force 380 generating function 1026 of inertia 436, 455, 457 of a sample 1046 vector 380 Monotone sequence 469 Moore's shortest path algorithm 960 Morera's theorem 661 Moulton 900 Moving trihedron (see Trihedron) M-test for convergence 969 Multinomial distribution 1025 Multiple point 391 Multiplication of complex numbers 603, 609 determinants 322 matrices 2]8,32l means 1038 power series I74,680 vectors 279,37I,37J Multiplication rule for events 1003 -] lndex Multiplicity 337,865 Multiply conněcted domain 646 Multistep method 898 "Multivalued function" 615 Mutually exclusive events 998 N Nabla 403 NAG 779 Natural frequency 63 logarithm 630, A60 spline 812 Neighborhood 387, 613 Nested form 786 NETLIB 779 Networks í32,146, 162, 244, 260, 263, 2'7], 33I in graph theory 973 Neumann, C. 20I functions 201 problem 558, 587,9I] Newton 14 -Cotes formulas 822 interpolation formulas 802, 805, 807 law of cooling 14 law of gravitation 385 method 800 -Raphson method 800 second law 62 Neyman 1049, 1058 Nicolson 924 Nicomedes 399 Nilpotent matrix 286 NIST 779 Nodal incidence matrix 27J Iine 514 Node I42,79] Nonbasic variables 945 Nonconservatíve 428 Nonhomogeneous differential equation 27, J8, 116, I59,305, 535 system of equations 288, 304 Nonlinear differential equations 45, 151 Nonorientable surface 453 Nonparametric test 1080 Nonsingular matrix 315 Norm 205, 326, 346, 359, 365, 849 Normal acceleration 395 asymptotically 1057 to a curve 398 derivative 444,465 distribution 1026, I047-I05], 1062-1067, A98 two-dimensional 1090 equations 860, 1086 form of a PDE 551 matrix 362,869 mode 542,582 plane 398 to a plane 375 random variable 1026 to a surface 44] vector 3]5,447 Nu1l hypothesis 1058 matrix (see Zero matrix) space 301 vector (see Zero vector) Nullity 301 Numeric methods'777 -934 differentiation 827 eigenvalues 863-882 equations 781-796 integration 811-821 interpolation 791-816 linear equations 833-858 matrix inversion 3I5, 844 optimization 936-953 ordinary differential equations (ODEs) 886_908 partial differential equations (PDEs) 909-930 Nystróm method 906 o o 962 Objective function 936 OC curve 1062 Odd function 490 ODE 4 (see also Differential equations) Ohm's law 92 one-dimensional heat equation 553 wave equation 539 One-sided derivative 484 test 1060 !l5 lndex One-step method 898 One-to-one mapping 729 Open disk 613 integration formula 827 interval ,{69 point set 613 Operating characteristic 1062 Operational calculus 59, 220 Operation count 838 Operator 59,32] Optimality principle, Bellman's 963 Optimal solution 942 Optimization 936-9 53, 9 59-990 Orbit 141 Order 887,962 of a determinant 308 of a differential equation 4, 535 of an iteration process 793 Ordering 969 Ordinary differential equation s 2-269, 886-908 (see also Differential equation) Orientable surface 452 orientation of a curve 638 surface 452 Orthogonal coordinates A7l curves 35 eigenvectors 350 expansion 210 functions 205,482 matrix 345 polynomials 209 series 210 trajectories 35 transformation 346 , vectors 326,37I Orthonormal functions 205, 2l0 oscillations of a beam 541,552 of a cable 198 in circuits 91 damped 64, 88 forced 84 free 61, 500,547 harmonic 63 of a mass on a spring 61,86, 135, I50,243, 252,261,342, 499 of a membrane 569-586 Oscillations (Cont.) self-sustained l57 of a string 204, 538,929 undamped 62, Osculating plane 398 Outcome 997 Outer product 377 Outlier 995 Output 26,230 Overdamping 65 Overdetermined sy stem 292 Overflow 782 Overrelaxation 851 Oveftone 542 P Paired comparison 1065 Pappus's theorem 458 Parabolic differential equation 55I, 922 Paraboloid 448 Parachutist 12 Parallelepiped 382 Parallel flow 766 Parallelogram equality 326,372, 612 Iaw 367 Parameter of a distribution 1016 Parametric representation 389, 446 Parking problem 1023 Parseval's equality 2I5, 504 Partial derivative 388, A66 differential equation 535, 909 fractions 23I,245 pivoting 29I,834 sum 171,480,666 Particular solution 6,48, 106, 159 Pascal 399 Path in a digraph 974 in a graph 959 of integration 42I, 637 PDE 535, 909 (see also Differential equation) Peaceman-Rachford method 915 Pearson, E. S. 1058 Pearson, K. 1066 Pendulum 68, l52, 156 Period 478 l16 lndex Periodic extension 494 function 478 Permutation 1006 Perron-Frobenius theorem 344, 869 Pfaffian form429 Phase angle 88 of complex number (see Argument) lag 88 plane I4I, I47 portrait I4I, I47 Picard iteration method 41 theorem 709 piecewise continuous 226 smooth 42I,448,639 Pivoting 291,834 Planar graph987 Plane 3I5,375 Plane curve 391 Poincaré 2í6 Point estimate 1046 at infinity 1I0,736 set 613 source 765 spectrum 507, 524 Poisson 769 distribution 1022, 1013, A91 equation 910, 918 integral formula 769 Polar coordinates 43J, 443, 580,607-608 form of complex numbers 607 moment on inertia 436 Pole 708 Polynomial approximation 797 matrix 865 Polynomially bounded 962 Polynomials 617 Chebyshev 209 Hermite 216 Laguerre 207,257 Legendre I79, 207, 590, 826 trigonometric 502 Population in statistics 1044 Population models 3I, I54,34I Position vector 366 positive definite 326, 312 Possible values 1012 Postman problem 963 Potential 40J, 42], 590, ]50, ]62 complex 763 theory 465, ]49 Power method for eigenvalues 872 of a test 1061 series 167, 673 series method 167 Precision 782 Predator-prey I54 Predictor-coffector 890, 900 Pre-Hilbert space 326 Prim's algorithm 971 Principal axes theorem 354 brarch 632 diagonal (see main diagonal) directions 340 normal (Fig.2I0) 397 part 708 value 60J, 630, 632, ]19,722 Prior estimate'794 Probability 1000, 1001 conditional 1003 density I0I4, 1034 distribution 1010, 1032 function I0I2, 1033 Producer's risk 1075 Product (see Multiplication) Projection of a vector 314 Pseudocode 183 Pure imaginary number 603 o QR-factorization method 879 Quadratic equation 785 form 353 interpolation ]99 Qualitative methods I24, 139-165 Quality control 1068 Quartile 995 Quasilinear 551, 909 Quotient of complex numbers 604 lndex R Rachford method 915 Radiation 7, 567 Radiocarbon dating 13 Radius of convergence 1]2, 675, 686 of a graph 973 Random experiment 991 numbers 1045 variable 1010. 1032 Range of a function 614 sample 994 Rank of a matrix 297,299,3I1 Raphson 790 Rational function 617 Ratio test 669 Rayleigh 159 equation 159 quotient 872 RC-circuit 97,23],240 Reactance 93 Real axis 604 part 602 sequence 664 vector space 324,369 Rectangular membrane 571 pulse 238,243 rule 817 wave 2II,480,488,492 Rectifiable curve 393 Rectification of a lot 1075 Rectifier 248,489,492 Rectifying plane 398 Reduction of order 50. 116 Region 433,614 Regression 1083 coefficient 1085. 1088 line 1084 Regula falsi 796 Regular point of an ODE 183 Sturm-Liouville problem 206 Rejectable quality level 1075 Rejection region 1060 Relative class frequency 994 Relative (Cont.) error J84 frequency 1000 Relaxation 850 Remainder I71,666 Removable singularity ]09 Representation 328 Residual 849,852 Residue 713 Residue theorem 715 Resistance 91 Resonance 86 Response 28,84 Restoring force 62 Resultant of forces 367 Riccati equation 34 Riemann 618 sphere 710 surface 746 Right-hand derivative 484 Iimit 484 Right-handed 371 Risk 1095 Rl-circuit 9],240 RLC-circuit 95, 24I, 244, 499 Robin problem 558, 587 Rodrigues's formula 181 Romberg integration 829 Root 610 Root test 671 Rotation 3B1, 385, 4I4,734,764 Rounding 782 Row echelon form294 -equivalent 292, 298 operations 292 scaling 838 space 300 sum norm 849 vector 274 Runge-Kutta-Fehlberg 893 Runge-Kutta methods 892, 9O4 Runge-Kutta-Nystróm 906 s Saddle point I43 Sample 997, 1045 covariance 1085 l17 lndex Sample (Cont.) Series (Cont.) distribution function 1076 hypergeometric 188 mean 996, 1045 infinite 666, A10 moments 1046 integration of 680 polnt 997 Laurent 70I, ]I2 range 994 Maclaurin 683 size 99], 1045 multiplication of I14, 680 space 997 of orthogonal functions 210 standard deviation 996, 1046 partial sums of I71,666 variance 996, IO45 power 167, 673 Sampling 1004, 1,023, 1073 real A69 SAS 991 remainder of IJI,666,684 Sawtooth wave 248,493,505 sum of 17I,666 Scalar 276,364 Taylor 683 field 384 trigonometrtc 4J9 function 384 value of I]I,666 matrtx 284 Serret-Frenet formulas multiplication 2]6,368 (see Frenet formulas) triple product 381 Set of points 613 Scaling 838 Shifted data problem 232 Scheduling 987 Shifting theorems 224,235, 528 Schoenberg 810 Shortest Schródinger 242 path 959 Schur's inequality 869 spanning tree 96'7 Schwartz 242 Shrine of the Book 814 Schwarz inequality 326 Sifttng 242 Secant 62'7, A62 Significance level 1059 methoď 794 Significant Second digit 781 Green's formula 466 in statistics 1059 shifting theorem 235 Sign test 1081 Sectionally continuous Similar matrices 350 (see Piecewise continuous) Simple Seidel 846 curve 391 Self-starting 898 event 997 Self-sustained oscillations 157 graph 955 Separable differential equation 12 pole 708 Separation of variables 12, 540 zero 709 Sequence 664, A69 Simplex Series 666, A69 method 944 addition of 680 table 945 of Bessel functions 2I3, 583 Simply connected 640, 646 binomial 689 Simpson's rule 821 convergence of I71,666 Simultaneous differentiation of I74,680 corrections 850 double Fourier 576 differential equations I24 of eigenfunctions 210 linear equations (see Linear systems) Fourier 2íI,480,487 Sine geometric 167, 668, 673, 687, 692 of a complex variable 627, 688,'742 harmonic 670 hyperbolic 688 lndex Sine (Cont.) integral 509, 690, A65, A95 of a real variable ,{60 Single precision 782 Single-valued relation 615 Singular at infinity 711 matrix 315 point I83, 686,70] solution 8, 50 Sturm-Liouville problem 206 Singularity 686,707 Sink 464, 765,913 SI system: Front cover Size of a sample 991, 1045 Skew-Hermitian 357, 36I Skewness 1020 Skew-symmetric matrix 283, 345 Skydiver 12 Slack variable 94I Slope field 9 Smooth curve 427,638 piecewise 42l, 448,639 surface 448 Sobolev 242 Soft spring I59 Software ]78,991 Solution of a differential equation 4, 46, I05, 536 general 6,48,106, 138 particular 6,48, 106 singular 8, 50 space 304 steady-state 88 of a system of differential equations 136 of a system of equations 288 vector 288 soR 851 Sorting 969,993 Source 464,765,973 Span 300 Spanning tree 967 Sparse graph 957 matrix 8I2,9I2 system of equations 846 Spectral density 520 mapping theorem 344, 865 Spectral (Cont.) radius 848 representation 520 shíft 344,865, 8]4 Spectrum 324, 542, 864 Speed 394 Sphere 446 Spherical coordinates 588, '{71 Spiral 399 point I44 Spline 811 S-PLUS, SPSS 992 Spring 62 Square error 503 matňx 214 membrane 575 root 792 wave 2l7,480,488,492 Stability 3I, 748, ]83,822,922 chart 148 Stagnation point 763 Staircase function 248 Standard basis 328,369 deviation 1016 form of a linear ODE 26, 45, I05 Standardized random variable 1018 Stationary point 937 statistical inference 1044 tables A96-A106 Steady 413,463 state 88 Steepest descent 938 Steiner 399,457 Stem-and-Ieaf ptot 994 Stencil 912 Step-by-step method 886 Step function 234 Step size control 889 Stereographic projection 71 1 Stiff oDE 896 system of ODEs 907 Stirling formula 1008, A64 stochastic matix 285 variable 1011 Stokes's theorem 469 lt9 I20 lndex Straight Itne 375,391 Stream functton 762 Streamline 16I Strength of a source ]6] Strictly diagonally dominant 868 String 204,538,594 Student's /-distribution 1053, A100 Sturm-Liouville problem 203 Subgraph 956 Submarine cable equations 594 Submatrix 302 Subsidiary equation 220, 230 Subspace 300 invariant 865 Success 1021 successive corrections B50 overrelaxation 851 Sum (see Addition) Sum of a series I71,666 Superlinear converg ence J 95 Superposition principle 106, 138 Surface 445 area 435, 442, 454 tntegral 449 normal 406 Surjective mapping 729 Symmetric matrix 283, 345 System of differential equations 124-165, 258-263, 902 linear equations (see Linear system) units: Front cover T Tables _ on differentiation: Front cover of Fourier transforms 529-531 of functions A94-A106 of integrals: Front cover of Laplace transforms 265-267 statistical A96-A106 Tangent 62], A62 to a curve 392,391 hyperbolic 629, A62 plane 406,447 vector 392 Tangential acceleration 395 Target 973 Tarjan 91I Taylor 683 formula 684 series 683 Tchebichef (s ee Chebyshev) r-distribution 1053, A100 Telegraph equations 594 Termination criterion 7 9I Termwise differentiation 696 integration 695 multiplication 680 Test chi-square 1077 for convergence 667 -672 of hypothesis 1058-1068 nonparametric 1080 Tetrahedron 382 Thermal conductivity 465,552 diffusivity 465,552 Three -eights rule 830 -sigma limits 1028 Timetabling 987 Torricelli's law 15 Torsion of a clxve 39J Torsional vibrations 68 Torus 454 Total differential 19 Trace of a matrix 344, 355,864 Trail 959 Traiectories 35, I33, I4I Transfer function 230 Transformation of Cartesian coordinates A84 by a complex function]29 of integrals 43], 439, 459, 469 Itnear 28l linear fracttonal734 orthogonal 346 similarity 350 unitary 359 of vector components A83 Transient state 88 Translation 365, ]34 Transmission line equations 593 Transpose of a matrix 282 Transpositions 1081, A106 Trapezoidal rule 817 Traveling salesman problem 960 lndex Tree 966 Trend 1081 Trial99] Triangle inequality 326,372, 608 Triangular matrix 283 Tricomi equation 55I, 552 Tridiag on altzation 8] 5 Tridiagonal matrix 8I2, 875, 9I4 Trigonometric approximation 502 form of complex numbers 60], 624 functions, complex 626, 688 functions, real A60 polynomial502 series 479 system 479,482 Trihedron 398 Triple integral 458 product 381 Trivial solution 2], 304 Truncation error 783 Tuning 543 Twisted curve 391 Two-dimensional distribution 1,032 normal distribution 1090 random variable 1032 wave equation 571 Two-sided test 1060 Type of a differential equation 551 Type I and II errors 1060 U UCL 1068 Unconstrained optimization 93] Unconelated 1090 Undamped system 62 Underdamping 66 Underdetermined sy stem 292 Underflow 782 Undetermined coefficients 79, II7, 160 Uniform convergence 69I distribution 1015, l0I7, 1034 Union of events 998 Uniqueness differential equations 37,73, 107, 13] Dirichlet problem774 Uniqueness (Cont.) Laurent series 705 linear equations 303 power series 678 Unit binormal vector 398 circle 611 impulse 242 matrtx 284 normal vector 44J principal normal vector 398 step function 234 tangent vector 392,398 vector 326 Unitary matrix 357 system of vectors 359 transformation 359 Unstable (see Stability) Upper control limit 1068 V Value of a series I]I,666 vandermonde determtnant Il2 Van der Pol equation 157 Variable complex 614 random 1010, 1032 standardized random 1018 stochastic 1011 variance of a distribution 1016 sample 996, 1045 Variation of parameters 98, 118, Vector 274,364 addition 276,327,367 field 384 function 384 moment 380 norm 853 product 377 space 300,323,369 subspace 300 Velocity 394 field 385 potentíal762 vecíor394 Venn diagram 998 Verhulst 31 l2l ii í l 160 lndex Vertex 955 coloring 987 exposed 9B3 incidence list 957 Vibration s (s ee Oscillations) Violin string 538 Vizing's theorem 987 yolta 92 Voltage drop 92 Volterra I54,20I,253 Volume 435 Yortex 767 Yorticity 764 W Waiting time problem 1013 Walk 959 Wave equation 536, 539, 569,929 Weber 2I7 eqlation 2I7 Weber (Cont.) functions 201 website see preface Weierstrass 618, 696 approximation theorem 797 M-test 696 Weighted graph 959 Weight function 205 Well-conditioned 852 Wessel 605 Wheatstone bridge 296 Work 313, 423 integral422 Wronskian 75, 108 z Zero of analytic function 709 matrix 276 vector 367 e- i:, e': Tť- ,,ll Ýi: some constants 2.] 1828 18284 59045 23536 1.64872 L2]0] 00128 14685 7.38905 60989 30650 22]23 3.I4I59 26535 89793 23846 9.86960 44010 89358 61883 I.71245 38509 05516 02130 logro rr : 0.49J 14 98]26 94133 85435 In rr : L1441298858 4g4OO 11414 1ogro e -- 0,43429 44819 03251 82765 ln 10 : 2.30258 50929 94045 68402 \/Ž : I.4t42I 35623 13095 04880 {2 : L25g9210498 94813 16411 Ý1 : t.]3205 08075 68811 29353 "Ý3 : I.44224g51O3 07408 38232 In2 : 0.6931411805 59945 30942 ln 3 : 1.09861 22886 68109 69140 y : 0.51121 56649 01,532 8606l In y : -0,54953 93129 81644 82234 (see Sec. 5.6) 1o : 0.01145 32925 19943 295]1 raď 1 rad : 51.29571 95130 82320 87680" : 5JoI7'44.806" polar coordinates x:rcos0 y--rsin0 , : \/r? + Ý tan 0 : L x dxdy: rdrd0 Series {m +-: ) _.* (|.r| < t) I-X rn:o ^- - š XllL e - ZJ ,rť. rn:l) ,ln": i (-I)*,2-*, m:o (2m -l l)t š (-I)*x2* cosx: Z CI*\| m:O co ,ffi m tn-l oo (-|)*xz^-I l l arctan.t : ž (l.rl m:o 2m* l '' ' 1) 1) Greek Alphabet ot Alpha P Beta y, l Gamma 6, A Delta , t Epsilon { Zeta T Eta 0, ů,a Theta c Iota K Kappa ^, A Lambda tL Mu uNu čXi o Omicron ŤPi p Rho cr, ž Sigma Ť Tau u, Y Upsilon ó, q, @ Phi x Chi Ý, Ý Psi @, d'L Omega Vectors a.b: arbr+ a2br* arb" |i j kl l"IaXb:la-l a2 asl | , "| lb, b2 bzl gradf : ví: #'- #: * #u r. F }ul óuz óus cllvV: V'V: - -r- - -r- - 0x óy 0z li j kl la a ól curlv:Vxo:l* ,, ,-l |r, t)2 ,, l