Theoretical Computer Science Cheat Sheet
Definitions Series
f(n) = O(g(n)) iff  positive c, n0 such that
0  f(n)  cg(n) n  n0.
n
i=1
i =
n(n + 1)
2
,
n
i=1
i2
=
n(n + 1)(2n + 1)
6
,
n
i=1
i3
=
n2
(n + 1)2
4
.
In general:
n
i=1
im
=
1
m + 1
(n + 1)m+1
- 1 -
n
i=1
(i + 1)m+1
- im+1
- (m + 1)im
n-1
i=1
im
=
1
m + 1
m
k=0
m + 1
k
Bknm+1-k
.
Geometric series:
n
i=0
ci
=
cn+1
- 1
c - 1
, c = 1,

i=0
ci
=
1
1 - c
,

i=1
ci
=
c
1 - c
, |c| < 1,
n
i=0
ici
=
ncn+2
- (n + 1)cn+1
+ c
(c - 1)2
, c = 1,

i=0
ici
=
c
(1 - c)2
, |c| < 1.
Harmonic series:
Hn =
n
i=1
1
i
,
n
i=1
iHi =
n(n + 1)
2
Hn n(n
- 1)
4
.
n
i=1
Hi = (n + 1)Hn - n,
n
i=1
i
m
Hi =
n + 1
m + 1
Hn+1 -
1
m + 1
.
f(n) = (g(n)) iff  positive c, n0 such that
f(n)  cg(n)  0 n  n0.
f(n) = (g(n)) iff f(n) = O(g(n)) and
f(n) = (g(n)).
f(n) = o(g(n)) iff limn f(n)/g(n) = 0.
lim
n
an = a iff  > 0, n0 such that
|an - a| < , n  n0.
sup S least b  R such that b  s,
s  S.
inf S greatest b  R such that b 
s, s  S.
lim inf
n
an lim
n
inf{ai | i  n, i  N}.
lim sup
n
an lim
n
sup{ai | i  n, i  N}.
n
k Combinations: Size k subsets
of a size n set.
n
k Stirling numbers (1st kind):
Arrangements of an n element
set into k cycles.
1.
n
k
=
n!
(n - k)!k!
, 2.
n
k=0
n
k
= 2n
, 3.
n
k
=
n
n - k
,
4.
n
k
=
n
k
n - 1
k - 1
, 5.
n
k
=
n - 1
k
+
n - 1
k - 1
,
6.
n
m
m
k
=
n
k
n - k
m - k
, 7.
n
k=0
r + k
k
=
r + n + 1
n
,
8.
n
k=0
k
m
=
n + 1
m + 1
, 9.
n
k=0
r
k
s
n - k
=
r + s
n
,
10.
n
k
= (-1)k k - n - 1
k
, 11.
n
1
=
n
n
= 1,
12.
n
2
= 2n-1
- 1, 13.
n
k
= k
n - 1
k
+
n - 1
k - 1
,
n
k Stirling numbers (2nd kind):
Partitions of an n element
set into k non-empty sets.
n
k 1st order Eulerian numbers:
Permutations 12 . . . n on
{1, 2, . . . , n} with k ascents.
n
k 2nd order Eulerian numbers.
Cn Catalan Numbers: Binary
trees with n + 1 vertices.
14.
n
1
= (n - 1)!, 15.
n
2
= (n - 1)!Hn-1, 16.
n
n
= 1, 17.
n
k

n
k
,
18.
n
k
= (n - 1)
n - 1
k
+
n - 1
k - 1
, 19.
n
n - 1
=
n
n - 1
=
n
2
, 20.
n
k=0
n
k
= n!, 21. Cn =
1
n + 1
2n
n
,
22.
n
0
=
n
n - 1
= 1, 23.
n
k
=
n
n - 1 - k
, 24.
n
k
= (k + 1)
n - 1
k
+ (n - k)
n - 1
k - 1
,
25.
0
k
=
1 if k = 0,
0 otherwise
26.
n
1
= 2n
- n - 1, 27.
n
2
= 3n
- (n + 1)2n
+
n + 1
2
,
28. xn
=
n
k=0
n
k
x + k
n
, 29.
n
m
=
m
k=0
n + 1
k
(m + 1 - k)n
(-1)k
, 30. m!
n
m
=
n
k=0
n
k
k
n - m
,
31.
n
m
=
n
k=0
n
k
n - k
m
(-1)n-k-m
k!, 32.
n
0
= 1, 33.
n
n
= 0 for n = 0,
34.
n
k
= (k + 1)
n - 1
k
+ (2n - 1 - k)
n - 1
k - 1
, 35.
n
k=0
n
k
=
(2n)n
2n
,
36.
x
x - n
=
n
k=0
n
k
x + n - 1 - k
2n
, 37.
n + 1
m + 1
=
k
n
k
k
m
=
n
k=0
k
m
(m + 1)n-k
,
Theoretical Computer Science Cheat Sheet
Identities Cont. Trees
38.
n + 1
m + 1
=
k
n
k
k
m
=
n
k=0
k
m
nn-k
= n!
n
k=0
1
k!
k
m
, 39.
x
x - n
=
n
k=0
n
k
x + k
2n
,
40.
n
m
=
k
n
k
k + 1
m + 1
(-1)n-k
, 41.
n
m
=
k
n + 1
k + 1
k
m
(-1)m-k
,
42.
m + n + 1
m
=
m
k=0
k
n + k
k
, 43.
m + n + 1
m
=
m
k=0
k(n + k)
n + k
k
,
44.
n
m
=
k
n + 1
k + 1
k
m
(-1)m-k
, 45. (n - m)!
n
m
=
k
n + 1
k + 1
k
m
(-1)m-k
, for n  m,
46.
n
n - m
=
k
m - n
m + k
m + n
n + k
m + k
k
, 47.
n
n - m
=
k
m - n
m + k
m + n
n + k
m + k
k
,
48.
n
+ m
+ m
=
k
k n - k
m
n
k
, 49.
n
+ m
+ m
=
k
k n - k
m
n
k
.
Every tree with n
vertices has n - 1
edges.
Kraft inequality:
If the depths
of the leaves of
a binary tree are
d1, . . . , dn:
n
i=1
2-di
 1,
and equality holds
only if every internal
node has 2
sons.
Recurrences
Master method:
T(n) = aT(n/b) + f(n), a  1, b > 1
If  > 0 such that f(n) = O(nlogb a-
)
then
T(n) = (nlogb a
).
If f(n) = (nlogb a
) then
T(n) = (nlogb a
log2 n).
If  > 0 such that f(n) = (nlogb a+
),
and c < 1 such that af(n/b)  cf(n)
for large n, then
T(n) = (f(n)).
Substitution (example): Consider the
following recurrence
Ti+1 = 22i
 T2
i , T1 = 2.
Note that Ti is always a power of two.
Let ti = log2 Ti. Then we have
ti+1 = 2i
+ 2ti, t1 = 1.
Let ui = ti/2i
. Dividing both sides of
the previous equation by 2i+1
we get
ti+1
2i+1
=
2i
2i+1
+
ti
2i
.
Substituting we find
ui+1 = 1
2 + ui, u1 = 1
2 ,
which is simply ui = i/2. So we find
that Ti has the closed form Ti = 2i2i-1
.
Summing factors (example): Consider
the following recurrence
T(n) = 3T(n/2) + n, T(1) = 1.
Rewrite so that all terms involving T
are on the left side
T(n) - 3T(n/2) = n.
Now expand the recurrence, and choose
a factor which makes the left side "tele-
scope"
1 T(n) - 3T(n/2) = n
3 T(n/2) - 3T(n/4) = n/2
...
...
...
3log2 n-1
T(2) - 3T(1) = 2
Let m = log2 n. Summing the left side
we get T(n) - 3m
T(1) = T(n) - 3m
=
T(n) - nk
where k = log2 3  1.58496.
Summing the right side we get
m-1
i=0
n
2i
3i
= n
m-1
i=0
3
2
i
.
Let c = 3
2 . Then we have
n
m-1
i=0
ci
= n
cm
- 1
c - 1
= 2n(clog2 n
- 1)
= 2n(c(k-1) logc n
- 1)
= 2nk
- 2n,
and so T(n) = 3nk
- 2n. Full history recurrences
can often be changed to limited
history ones (example): Consider
Ti = 1 +
i-1
j=0
Tj, T0 = 1.
Note that
Ti+1 = 1 +
i
j=0
Tj.
Subtracting we find
Ti+1 - Ti = 1 +
i
j=0
Tj - 1 -
i-1
j=0
Tj
= Ti.
And so Ti+1 = 2Ti = 2i+1
.
Generating functions:
1. Multiply both sides of the equation
by xi
.
2. Sum both sides over all i for
which the equation is valid.
3. Choose a generating function
G(x). Usually G(x) =

i=0 xi
gi.
3. Rewrite the equation in terms of
the generating function G(x).
4. Solve for G(x).
5. The coefficient of xi
in G(x) is gi.
Example:
gi+1 = 2gi + 1, g0 = 0.
Multiply and sum:
i0
gi+1xi
=
i0
2gixi
+
i0
xi
.
We choose G(x) = i0 xi
gi. Rewrite
in terms of G(x):
G(x) - g0
x
= 2G(x) +
i0
xi
.
Simplify:
G(x)
x
= 2G(x) +
1
1 - x
.
Solve for G(x):
G(x) =
x
(1 - x)(1 - 2x)
.
Expand this using partial fractions:
G(x) = x
2
1 - 2x
-
1
1 - x
= x

2
i0
2i
xi
-
i0
xi


=
i0
(2i+1
- 1)xi+1
.
So gi = 2i
- 1.
Theoretical Computer Science Cheat Sheet
  3.14159, e  2.71828,   0.57721,  = 1+

5
2  1.61803, ^ = 1-

5
2  -.61803
i 2i
pi General Probability
1 2 2 Bernoulli Numbers (Bi = 0, odd i = 1):
B0 = 1, B1 = -1
2 , B2 = 1
6 , B4 = - 1
30 ,
B6 = 1
42 , B8 = - 1
30 , B10 = 5
66 .
Change of base, quadratic formula:
logb x =
loga x
loga b
,
-b 

b2 - 4ac
2a
.
Euler's number e:
e = 1 + 1
2 + 1
6 + 1
24 + 1
120 +   
lim
n
1 +
x
n
n
= ex
.
1 + 1
n
n
< e < 1 + 1
n
n+1
.
1 + 1
n
n
= e -
e
2n
+
11e
24n2
- O
1
n3
.
Harmonic numbers:
1, 3
2 , 11
6 , 25
12 , 137
60 , 49
20 , 363
140 , 761
280 , 7129
2520 , . . .
ln n < Hn < ln n + 1,
Hn = ln n +  + O
1
n
.
Factorial, Stirling's approximation:
1, 2, 6, 24, 120, 720, 5040, 40320, 362880, . . .
n! =

2n
n
e
n
1 + 
1
n
.
Ackermann's function and inverse:
a(i, j) =



2j
i = 1
a(i - 1, 2) j = 1
a(i - 1, a(i, j - 1)) i, j  2
(i) = min{j | a(j, j)  i}.
Continuous distributions: If
Pr[a < X < b] =
b
a
p(x) dx,
then p is the probability density function of
X. If
Pr[X < a] = P(a),
then P is the distribution function of X. If
P and p both exist then
P(a) =
a
p(x)
dx.
Expectation: If X is discrete
E[g(X)] =
x
g(x) Pr[X = x].
If X continuous then
E[g(X)] =

g(x)p(x)
dx =

g(x)
dP(x).
Variance, standard deviation:
VAR[X] = E[X2
] - E[X]2
,
 = VAR[X].
For events A and B:
Pr[A  B] = Pr[A] + Pr[B] - Pr[A  B]
Pr[A  B] = Pr[A]  Pr[B],
iff A and B are independent.
Pr[A|B] =
Pr[A  B]
Pr[B]
For random variables X and Y :
E[X  Y ] = E[X]  E[Y ],
if X and Y are independent.
E[X + Y ] = E[X] + E[Y ],
E[cX] = c E[X].
Bayes' theorem:
Pr[Ai|B] =
Pr[B|Ai] Pr[Ai]
n
j=1 Pr[Aj] Pr[B|Aj]
.
Inclusion-exclusion:
Pr
n
i=1
Xi =
n
i=1
Pr[Xi] +
n
k=2
(-1)k+1
ii<<ik
Pr
k
j=1
Xij
.
Moment inequalities:
Pr |X|   E[X] 
1

,
Pr X - E[X]     
1
2
.
Geometric distribution:
Pr[X = k] = pqk-1
, q = 1 - p,
E[X] =

k=1
kpqk-1
=
1
p
.
2 4 3
3 8 5
4 16 7
5 32 11
6 64 13
7 128 17
8 256 19
9 512 23
10 1,024 29
11 2,048 31
12 4,096 37
13 8,192 41
14 16,384 43
15 32,768 47
16 65,536 53
17 131,072 59
18 262,144 61
19 524,288 67
20 1,048,576 71
21 2,097,152 73
22 4,194,304 79
23 8,388,608 83
24 16,777,216 89
25 33,554,432 97
26 67,108,864 101
27 134,217,728 103
28 268,435,456 107 Binomial distribution:
Pr[X = k] =
n
k
pk
qn-k
, q = 1 - p,
E[X] =
n
k=1
k
n
k
pk
qn-k
= np.
Poisson distribution:
Pr[X = k] =
e-
k
k!
, E[X] = .
Normal (Gaussian) distribution:
p(x) =
1

2
e-(x-)2
/22
, E[X] = .
The "coupon collector": We are given a
random coupon each day, and there are n
different types of coupons. The distribution
of coupons is uniform. The expected
number of days to pass before we to collect
all n types is
nHn.
29 536,870,912 109
30 1,073,741,824 113
31 2,147,483,648 127
32 4,294,967,296 131
Pascaľs Triangle
1
1 1
1 2 1
1 3 3 1
1 4 6 4 1
1 5 10 10 5 1
1 6 15 20 15 6 1
1 7 21 35 35 21 7 1
1 8 28 56 70 56 28 8 1
1 9 36 84 126 126 84 36 9 1
1 10 45 120 210 252 210 120 45 10 1
Theoretical Computer Science Cheat Sheet
Trigonometry Matrices More Trig.
A
c

B
a
b
C
(0,-1)
(0,1)
(-1,0) (1,0)
(cos , sin )
Pythagorean theorem:
C2
= A2
+ B2
.
Definitions:
sin a = A/C, cos a = B/C,
csc a = C/A, sec a = C/B,
tan a =
sin a
cos a
=
A
B
, cot a =
cos a
sin a
=
B
A
.
Area, radius of inscribed circle:
1
2 AB,
AB
A + B + C
.
Identities:
sin x =
1
csc x
, cos x =
1
sec x
,
tan x =
1
cot x
, sin2
x + cos2
x = 1,
1 + tan2
x = sec2
x, 1 + cot2
x = csc2
x,
sin x = cos 
2 - x , sin x = sin( - x),
cos x = - cos( - x), tan x = cot 
2 - x ,
cot x = - cot( - x), csc x = cot x
2 - cot x,
sin(x  y) = sin x cos y  cos x sin y,
cos(x  y) = cos x cos y sin x sin y,
tan(x  y) =
tan x  tan y
1 tan x tan y
,
cot(x  y) =
cot x cot y 1
cot x  cot y
,
sin 2x = 2 sin x cos x, sin 2x =
2 tan x
1 + tan2
x
,
cos 2x = cos2
x - sin2
x, cos 2x = 2 cos2
x - 1,
cos 2x = 1 - 2 sin2
x, cos 2x =
1 - tan2
x
1 + tan2
x
,
tan 2x =
2 tan x
1 - tan2
x
, cot 2x =
cot2
x - 1
2 cot x
,
sin(x + y) sin(x - y) = sin2
x - sin2
y,
cos(x + y) cos(x - y) = cos2
x - sin2
y.
Euler's equation:
eix
= cos x + i sin x, ei
= -1.
Multiplication:
C = A  B, ci,j =
n
k=1
ai,kbk,j.
Determinants: det A = 0 iff A is non-singular.
det A  B = det A  det B,
det A =

n
i=1
sign()ai,(i).
2 × 2 and 3 × 3 determinant:
a b
c d
= ad - bc,
a b c
d e f
g h i
= g
b c
e f
- h
a c
d f
+ i
a b
d e
=
aei + bfg + cdh
- ceg - fha - ibd.
Permanents:
perm A =

n
i=1
ai,(i).
A
a
c
hb
B
C
Law of cosines:
c2
= a2
+b2
-2ab cos C.
Area:
A = 1
2 hc,
= 1
2 ab sin C,
=
c2
sin A sin B
2 sin C
.
Heron's formula:
A =

s  sa  sb  sc,
s = 1
2 (a + b + c),
sa = s - a,
sb = s - b,
sc = s - c.
More identities:
sin x
2 =
1 - cos x
2
,
cos x
2 =
1 + cos x
2
,
tan x
2 =
1 - cos x
1 + cos x
,
=
1 - cos x
sin x
,
=
sin x
1 + cos x
,
cot x
2 =
1 + cos x
1 - cos x
,
=
1 + cos x
sin x
,
=
sin x
1 - cos x
,
sin x =
eix
- e-ix
2i
,
cos x =
eix
+ e-ix
2
,
tan x = -i
eix
- e-ix
eix + e-ix
,
= -i
e2ix
- 1
e2ix + 1
,
sin x =
sinh ix
i
,
cos x = cosh ix,
tan x =
tanh ix
i
.
Hyperbolic Functions
Definitions:
sinh x =
ex
- e-x
2
, cosh x =
ex
+ e-x
2
,
tanh x =
ex
- e-x
ex + e-x
, csch x =
1
sinh x
,
sech x =
1
cosh x
, coth x =
1
tanh x
.
Identities:
cosh2
x - sinh2
x = 1, tanh2
x + sech2
x = 1,
coth2
x - csch2
x = 1, sinh(-x) = - sinh x,
cosh(-x) = cosh x, tanh(-x) = - tanh x,
sinh(x + y) = sinh x cosh y + cosh x sinh y,
cosh(x + y) = cosh x cosh y + sinh x sinh y,
sinh 2x = 2 sinh x cosh x,
cosh 2x = cosh2
x + sinh2
x,
cosh x + sinh x = ex
, cosh x - sinh x = e-x
,
(cosh x + sinh x)n
= cosh nx + sinh nx, n  Z,
2 sinh2 x
2 = cosh x - 1, 2 cosh2 x
2 = cosh x + 1.
 sin  cos  tan 
0 0 1 0

6
1
2

3
2

3
3

4

2
2

2
2 1

3

3
2
1
2

3

2 1 0 
. . . in mathematics
you don't understand
things, you
just get used to
them.
­ J. von Neumannv2.02 c 1994 by Steve Seiden
sseiden@acm.org
http://www.csc.lsu.edu/~seiden
Theoretical Computer Science Cheat Sheet
Number Theory Graph Theory
The Chinese remainder theorem: There exists
a number C such that:
C  r1 mod m1
...
...
...
C  rn mod mn
if mi and mj are relatively prime for i = j.
Euler's function: (x) is the number of
positive integers less than x relatively
prime to x. If
n
i=1 pei
i is the prime factorization
of x then
(x) =
n
i=1
pei-1
i (pi - 1).
Euler's theorem: If a and b are relatively
prime then
1  a(b)
mod b.
Fermaťs theorem:
1  ap-1
mod p.
The Euclidean algorithm: if a > b are integers
then
gcd(a, b) = gcd(a mod b, b).
If
n
i=1 pei
i is the prime factorization of x
then
S(x) =
d|x
d =
n
i=1
pei+1
i - 1
pi - 1
.
Perfect Numbers: x is an even perfect number
iff x = 2n-1
(2n
-1) and 2n
-1 is prime.
Wilson's theorem: n is a prime iff
(n - 1)!  -1 mod n.
M¨obius inversion:
(i) =



1 if i = 1.
0 if i is not square-free.
(-1)r
if i is the product of
r distinct primes.
If
G(a) =
d|a
F(d),
then
F(a) =
d|a
(d)G
a
d
.
Prime numbers:
pn = n ln n + n ln ln n - n + n
ln ln n
ln n
+ O
n
ln n
,
(n) =
n
ln n
+
n
(ln n)2
+
2!n
(ln n)3
+ O
n
(ln n)4
.
Definitions:
Loop An edge connecting a vertex
to itself.
Directed Each edge has a direction.
Simple Graph with no loops or
multi-edges.
Walk A sequence v0e1v1 . . . e v .
Trail A walk with distinct edges.
Path A trail with distinct
vertices.
Connected A graph where there exists
a path between any two
vertices.
Component A maximal connected
subgraph.
Tree A connected acyclic graph.
Free tree A tree with no root.
DAG Directed acyclic graph.
Eulerian Graph with a trail visiting
each edge exactly once.
Hamiltonian Graph with a cycle visiting
each vertex exactly once.
Cut A set of edges whose removal
increases the number
of components.
Cut-set A minimal cut.
Cut edge A size 1 cut.
k-Connected A graph connected with
the removal of any k - 1
vertices.
k-Tough S  V, S =  we have
k  c(G - S)  |S|.
k-Regular A graph where all vertices
have degree k.
k-Factor A k-regular spanning
subgraph.
Matching A set of edges, no two of
which are adjacent.
Clique A set of vertices, all of
which are adjacent.
Ind. set A set of vertices, none of
which are adjacent.
Vertex cover A set of vertices which
cover all edges.
Planar graph A graph which can be embeded
in the plane.
Plane graph An embedding of a planar
graph.
vV
deg(v) = 2m.
If G is planar then n - m + f = 2, so
f  2n - 4, m  3n - 6.
Any planar graph has a vertex with degree
 5.
Notation:
E(G) Edge set
V (G) Vertex set
c(G) Number of components
G[S] Induced subgraph
deg(v) Degree of v
(G) Maximum degree
(G) Minimum degree
(G) Chromatic number
E(G) Edge chromatic number
Gc
Complement graph
Kn Complete graph
Kn1,n2
Complete bipartite graph
r(k, ) Ramsey number
Geometry
Projective coordinates: triples
(x, y, z), not all x, y and z zero.
(x, y, z) = (cx, cy, cz) c = 0.
Cartesian Projective
(x, y) (x, y, 1)
y = mx + b (m, -1, b)
x = c (1, 0, -c)
Distance formula, Lp and L
metric:
(x1 - x0)2 + (y1 - y0)2,
|x1 - x0|p
+ |y1 - y0|p 1/p
,
lim
p
|x1 - x0|p
+ |y1 - y0|p 1/p
.
Area of triangle (x0, y0), (x1, y1)
and (x2, y2):
1
2 abs
x1 - x0 y1 - y0
x2 - x0 y2 - y0
.
Angle formed by three points:

(0, 0) (x1, y1)
(x2, y2)
2
1
cos  =
(x1, y1)  (x2, y2)
1 2
.
Line through two points (x0, y0)
and (x1, y1):
x y 1
x0 y0 1
x1 y1 1
= 0.
Area of circle, volume of sphere:
A = r2
, V = 4
3 r3
.
If I have seen farther than others,
it is because I have stood on the
shoulders of giants.
­ Issac Newton
Theoretical Computer Science Cheat Sheet
 Calculus
Wallis' identity:
 = 2 
2  2  4  4  6  6   
1  3  3  5  5  7   
Brouncker's continued fraction expansion:

4 = 1 +
12
2 + 32
2+ 52
2+ 72
2+
Gregrory's series:

4 = 1 - 1
3 + 1
5 - 1
7 + 1
9 -   
Newton's series:

6 =
1
2
+
1
2  3  23
+
1  3
2  4  5  25
+   
Sharp's series:

6 =
1

3
1-
1
31  3
+
1
32  5
-
1
33  7
+  
Euler's series:
2
6 = 1
12 + 1
22 + 1
32 + 1
42 + 1
52 +   
2
8 = 1
12 + 1
32 + 1
52 + 1
72 + 1
92 +   
2
12 = 1
12 - 1
22 + 1
32 - 1
42 + 1
52 -   
Derivatives:
1.
d(cu)
dx
= c
du
dx
, 2.
d(u + v)
dx
=
du
dx
+
dv
dx
, 3.
d(uv)
dx
= u
dv
dx
+ v
du
dx
,
4.
d(un
)
dx
= nun-1 du
dx
, 5.
d(u/v)
dx
=
v du
dx - u dv
dx
v2
, 6.
d(ecu
)
dx
= cecu du
dx
,
7.
d(cu
)
dx
= (ln c)cu du
dx
, 8.
d(ln u)
dx
=
1
u
du
dx
,
9.
d(sin u)
dx
= cos u
du
dx
, 10.
d(cos u)
dx
= - sin u
du
dx
,
11.
d(tan u)
dx
= sec2
u
du
dx
, 12.
d(cot u)
dx
= csc2
u
du
dx
,
13.
d(sec u)
dx
= tan u sec u
du
dx
, 14.
d(csc u)
dx
= - cot u csc u
du
dx
,
15.
d(arcsin u)
dx
=
1

1 - u2
du
dx
, 16.
d(arccos u)
dx
=
-1

1 - u2
du
dx
,
17.
d(arctan u)
dx
=
1
1 + u2
du
dx
, 18.
d(arccot u)
dx
=
-1
1 + u2
du
dx
,
19.
d(arcsec u)
dx
=
1
u

1 - u2
du
dx
, 20.
d(arccsc u)
dx
=
-1
u

1 - u2
du
dx
,
21.
d(sinh u)
dx
= cosh u
du
dx
, 22.
d(cosh u)
dx
= sinh u
du
dx
,
23.
d(tanh u)
dx
= sech2
u
du
dx
, 24.
d(coth u)
dx
= - csch2
u
du
dx
,
25.
d(sech u)
dx
= - sech u tanh u
du
dx
, 26.
d(csch u)
dx
= - csch u coth u
du
dx
,
27.
d(arcsinh u)
dx
=
1

1 + u2
du
dx
, 28.
d(arccosh u)
dx
=
1

u2 - 1
du
dx
,
29.
d(arctanh u)
dx
=
1
1 - u2
du
dx
, 30.
d(arccoth u)
dx
=
1
u2 - 1
du
dx
,
31.
d(arcsech u)
dx
=
-1
u

1 - u2
du
dx
, 32.
d(arccsch u)
dx
=
-1
|u|

1 + u2
du
dx
.
Integrals:
1. cu dx = c u dx, 2. (u + v) dx = u dx + v dx,
3. xn
dx =
1
n + 1
xn+1
, n = -1, 4.
1
x
dx = ln x, 5. ex
dx = ex
,
6.
dx
1 + x2
= arctan x, 7. u
dv
dx
dx = uv - v
du
dx
dx,
8. sin x dx = - cos x, 9. cos x dx = sin x,
10. tan x dx = - ln | cos x|, 11. cot x dx = ln | cos x|,
12. sec x dx = ln | sec x + tan x|, 13. csc x dx = ln | csc x + cot x|,
14. arcsin x
a dx = arcsin x
a + a2 - x2, a > 0,
Partial Fractions
Let N(x) and D(x) be polynomial functions
of x. We can break down
N(x)/D(x) using partial fraction expansion.
First, if the degree of N is greater
than or equal to the degree of D, divide
N by D, obtaining
N(x)
D(x)
= Q(x) +
N (x)
D(x)
,
where the degree of N is less than that of
D. Second, factor D(x). Use the following
rules: For a non-repeated factor:
N(x)
(x - a)D(x)
=
A
x - a
+
N (x)
D(x)
,
where
A =
N(x)
D(x) x=a
.
For a repeated factor:
N(x)
(x - a)mD(x)
=
m-1
k=0
Ak
(x - a)m-k
+
N (x)
D(x)
,
where
Ak =
1
k!
dk
dxk
N(x)
D(x) x=a
.
The reasonable man adapts himself to the
world; the unreasonable persists in trying
to adapt the world to himself. Therefore
all progress depends on the unreasonable.
­ George Bernard Shaw
Theoretical Computer Science Cheat Sheet
Calculus Cont.
15. arccos x
a dx = arccos x
a - a2 - x2, a > 0, 16. arctan x
a dx = x arctan x
a - a
2 ln(a2
+ x2
), a > 0,
17. sin2
(ax)dx = 1
2a ax - sin(ax) cos(ax) , 18. cos2
(ax)dx = 1
2a ax + sin(ax) cos(ax) ,
19. sec2
x dx = tan x, 20. csc2
x dx = - cot x,
21. sinn
x dx = -
sinn-1
x cos x
n
+
n - 1
n
sinn-2
x dx, 22. cosn
x dx =
cosn-1
x sin x
n
+
n - 1
n
cosn-2
x dx,
23. tann
x dx =
tann-1
x
n - 1
- tann-2
x dx, n = 1, 24. cotn
x dx = -
cotn-1
x
n - 1
- cotn-2
x dx, n = 1,
25. secn
x dx =
tan x secn-1
x
n - 1
+
n - 2
n - 1
secn-2
x dx, n = 1,
26. cscn
x dx = cot
x cscn-1
x
n - 1
+
n - 2
n - 1
cscn-2
x dx, n = 1, 27. sinh x dx = cosh x, 28. cosh x dx = sinh x,
29. tanh x dx = ln | cosh x|, 30. coth x dx = ln | sinh x|, 31. sech x dx = arctan sinh x, 32. csch x dx = ln tanh x
2 ,
33. sinh2
x dx = 1
4 sinh(2x) - 1
2 x, 34. cosh2
x dx = 1
4 sinh(2x) + 1
2 x, 35. sech2
x dx = tanh x,
36. arcsinh x
a dx = x arcsinh x
a - x2 + a2, a > 0, 37. arctanh x
a dx = x arctanh x
a + a
2 ln |a2
- x2
|,
38. arccosh x
a dx =



x arccosh
x
a
- x2 + a2, if arccosh x
a > 0 and a > 0,
x arccosh
x
a
+ x2 + a2, if arccosh x
a < 0 and a > 0,
39.
dx

a2 + x2
= ln x + a2 + x2 , a > 0,
40.
dx
a2 + x2
= 1
a arctan x
a , a > 0, 41. a2 - x2 dx = x
2 a2 - x2 + a2
2 arcsin x
a , a > 0,
42. (a2
- x2
)3/2
dx = x
8 (5a2
- 2x2
) a2 - x2 + 3a4
8 arcsin x
a , a > 0,
43.
dx

a2 - x2
= arcsin x
a , a > 0, 44.
dx
a2 - x2
=
1
2a
ln
a + x
a - x
, 45.
dx
(a2 - x2)3/2
=
x
a2

a2 - x2
,
46. a2  x2 dx = x
2 a2  x2  a2
2 ln x + a2  x2 , 47.
dx

x2 - a2
= ln x + x2 - a2 , a > 0,
48.
dx
ax2 + bx
=
1
a
ln
x
a + bx
, 49. x

a + bx dx =
2(3bx - 2a)(a + bx)3/2
15b2
,
50.

a + bx
x
dx = 2

a + bx + a
1
x

a + bx
dx, 51.
x

a + bx
dx =
1

2
ln

a + bx -

a

a + bx +

a
, a > 0,
52.

a2 - x2
x
dx = a2 - x2 - a ln
a +

a2 - x2
x
, 53. x a2 - x2 dx = -1
3 (a2
- x2
)3/2
,
54. x2
a2 - x2 dx = x
8 (2x2
- a2
) a2 - x2 + a4
8 arcsin x
a , a > 0, 55.
dx

a2 - x2
= -1
a ln
a +

a2 - x2
x
,
56.
x dx

a2 - x2
= - a2 - x2, 57.
x2
dx

a2 - x2
= -x
2 a2 - x2 + a2
2 arcsin x
a, a > 0,
58.

a2 + x2
x
dx = a2 + x2 - a ln
a +

a2 + x2
x
, 59.

x2 - a2
x
dx = x2 - a2 - a arccos a
|x| , a > 0,
60. x x2  a2 dx = 1
3 (x2
 a2
)3/2
, 61.
dx
x

x2 + a2
= 1
a ln
x
a +

a2 + x2
,
Theoretical Computer Science Cheat Sheet
Calculus Cont. Finite Calculus
62.
dx
x

x2 - a2
= 1
a arccos a
|x| , a > 0, 63.
dx
x2

x2  a2
=

x2  a2
a2x
,
64.
x dx

x2  a2
= x2  a2, 65.

x2  a2
x4
dx =
(x2
+ a2
)3/2
3a2x3
,
66.
dx
ax2 + bx + c
=



1

b2 - 4ac
ln
2ax + b -

b2 - 4ac
2ax + b +

b2 - 4ac
, if b2
> 4ac,
2

4ac - b2
arctan
2ax + b

4ac - b2
, if b2
< 4ac,
67.
dx

ax2 + bx + c
=



1

a
ln 2ax + b + 2

a ax2 + bx + c , if a > 0,
1

-a
arcsin
-2ax - b

b2 - 4ac
, if a < 0,
68. ax2 + bx + c dx =
2ax + b
4a
ax2 + bx + c +
4ax - b2
8a
dx

ax2 + bx + c
,
69.
x dx

ax2 + bx + c
=

ax2 + bx + c
a
-
b
2a
dx

ax2 + bx + c
,
70.
dx
x

ax2 + bx + c
=



-1

c
ln
2

c

ax2 + bx + c + bx + 2c
x
, if c > 0,
1

-c
arcsin
bx + 2c
|x|

b2 - 4ac
, if c < 0,
71. x3
x2 + a2 dx = (1
3 x2
- 2
15 a2
)(x2
+ a2
)3/2
,
72. xn
sin(ax) dx = -1
a xn
cos(ax) + n
a xn-1
cos(ax) dx,
73. xn
cos(ax) dx = 1
a xn
sin(ax) - n
a xn-1
sin(ax) dx,
74. xn
eax
dx =
xn
eax
a
- n
a xn-1
eax
dx,
75. xn
ln(ax) dx = xn+1 ln(ax)
n + 1
-
1
(n + 1)2
,
76. xn
(ln ax)m
dx =
xn+1
n + 1
(ln ax)m
-
m
n + 1
xn
(ln ax)m-1
dx.
Difference, shift operators:
f(x) = f(x + 1) - f(x),
E f(x) = f(x + 1).
Fundamental Theorem:
f(x) = F(x)  f(x)x = F(x) + C.
b
a
f(x)x =
b-1
i=a
f(i).
Differences:
(cu) = cu, (u + v) = u + v,
(uv) = uv + E vu,
(xn
) = nxn-1
,
(Hx) = x-1
, (2x
) = 2x
,
(cx
) = (c - 1)cx
,  x
m = x
m-1 .
Sums:
cu x = c u x,
(u + v) x = u x + v x,
uv x = uv - E vu x,
xn
x = x
n+1
m+1 , x-1
x = Hx,
cx
x = cx
c-1 , x
m x = x
m+1 .
Falling Factorial Powers:
xn
= x(x - 1)    (x - n + 1), n > 0,
x0
= 1,
xn
=
1
(x + 1)    (x + |n|)
, n < 0,
xn+m
= xm
(x - m)n
.
Rising Factorial Powers:
xn
= x(x + 1)    (x + n - 1), n > 0,
x0
= 1,
xn
=
1
(x - 1)    (x - |n|)
, n < 0,
xn+m
= xm
(x + m)n
.
Conversion:
xn
= (-1)n
(-x)n
= (x - n + 1)n
= 1/(x + 1)-n
,
xn
= (-1)n
(-x)n
= (x + n - 1)n
= 1/(x - 1)-n
,
xn
=
n
k=1
n
k
xk
=
n
k=1
n
k
(-1)n-k
xk
,
xn
=
n
k=1
n
k
(-1)n-k
xk
,
xn
=
n
k=1
n
k
xk
.
x1
= x1
= x1
x2
= x2
+ x1
= x2
- x1
x3
= x3
+ 3x2
+ x1
= x3
- 3x2
+ x1
x4
= x4
+ 6x3
+ 7x2
+ x1
= x4
- 6x3
+ 7x2
- x1
x5
= x5
+ 15x4
+ 25x3
+ 10x2
+ x1
= x5
- 15x4
+ 25x3
- 10x2
+ x1
x1
= x1
x1
= x1
x2
= x2
+ x1
x2
= x2
- x1
x3
= x3
+ 3x2
+ 2x1
x3
= x3
- 3x2
+ 2x1
x4
= x4
+ 6x3
+ 11x2
+ 6x1
x4
= x4
- 6x3
+ 11x2
- 6x1
x5
= x5
+ 10x4
+ 35x3
+ 50x2
+ 24x1
x5
= x5
- 10x4
+ 35x3
- 50x2
+ 24x1
Theoretical Computer Science Cheat Sheet
Series
Taylor's series:
f(x) = f(a) + (x - a)f (a) +
(x - a)2
2
f (a) +    =

i=0
(x - a)i
i!
f(i)
(a).
Expansions:
1
1 - x
= 1 + x + x2
+ x3
+ x4
+    =

i=0
xi
,
1
1 - cx
= 1 + cx + c2
x2
+ c3
x3
+    =

i=0
ci
xi
,
1
1 - xn
= 1 + xn
+ x2n
+ x3n
+    =

i=0
xni
,
x
(1 - x)2
= x + 2x2
+ 3x3
+ 4x4
+    =

i=0
ixi
,
xk dn
dxn
1
1 - x
= x + 2n
x2
+ 3n
x3
+ 4n
x4
+    =

i=0
in
xi
,
ex
= 1 + x + 1
2 x2
+ 1
6 x3
+    =

i=0
xi
i!
,
ln(1 + x) = x - 1
2 x2
+ 1
3 x3
- 1
4 x4
-    =

i=1
(-1)i+1 xi
i
,
ln
1
1 - x
= x + 1
2 x2
+ 1
3 x3
+ 1
4 x4
+    =

i=1
xi
i
,
sin x = x - 1
3! x3
+ 1
5! x5
- 1
7! x7
+    =

i=0
(-1)i x2i+1
(2i + 1)!
,
cos x = 1 - 1
2! x2
+ 1
4! x4
- 1
6! x6
+    =

i=0
(-1)i x2i
(2i)!
,
tan-1
x = x - 1
3 x3
+ 1
5 x5
- 1
7 x7
+    =

i=0
(-1)i x2i+1
(2i + 1)
,
(1 + x)n
= 1 + nx + n(n-1)
2 x2
+    =

i=0
n
i
xi
,
1
(1 - x)n+1
= 1 + (n + 1)x + n+2
2 x2
+    =

i=0
i + n
i
xi
,
x
ex - 1
= 1 - 1
2 x + 1
12 x2
- 1
720 x4
+    =

i=0
Bixi
i!
,
1
2x
(1 -

1 - 4x) = 1 + x + 2x2
+ 5x3
+    =

i=0
1
i + 1
2i
i
xi
,
1

1 - 4x
= 1 + x + 2x2
+ 6x3
+    =

i=0
2i
i
xi
,
1

1 - 4x
1 -

1 - 4x
2x
n
= 1 + (2 + n)x + 4+n
2 x2
+    =

i=0
2i + n
i
xi
,
1
1 - x
ln
1
1 - x
= x + 3
2 x2
+ 11
6 x3
+ 25
12 x4
+    =

i=1
Hixi
,
1
2
ln
1
1 - x
2
= 1
2 x2
+ 3
4 x3
+ 11
24 x4
+    =

i=2
Hi-1xi
i
,
x
1 - x - x2
= x + x2
+ 2x3
+ 3x4
+    =

i=0
Fixi
,
Fnx
1 - (Fn-1 + Fn+1)x - (-1)nx2
= Fnx + F2nx2
+ F3nx3
+    =

i=0
Fnixi
.
Ordinary power series:
A(x) =

i=0
aixi
.
Exponential power series:
A(x) =

i=0
ai
xi
i!
.
Dirichlet power series:
A(x) =

i=1
ai
ix
.
Binomial theorem:
(x + y)n
=
n
k=0
n
k
xn-k
yk
.
Difference of like powers:
xn
- yn
= (x - y)
n-1
k=0
xn-1-k
yk
.
For ordinary power series:
A(x) + B(x) =

i=0
(ai + bi)xi
,
xk
A(x) =

i=k
ai-kxi
,
A(x) -
k-1
i=0 aixi
xk
=

i=0
ai+kxi
,
A(cx) =

i=0
ci
aixi
,
A (x) =

i=0
(i + 1)ai+1xi
,
xA (x) =

i=1
iaixi
,
A(x) dx =

i=1
ai-1
i
xi
,
A(x) + A(-x)
2
=

i=0
a2ix2i
,
A(x) - A(-x)
2
=

i=0
a2i+1x2i+1
.
Summation: If bi =
i
j=0 ai then
B(x) =
1
1 - x
A(x).
Convolution:
A(x)B(x) =

i=0


i
j=0
ajbi-j

 xi
.
God made the natural numbers;
all the rest is the work of man.
­ Leopold Kronecker
Theoretical Computer Science Cheat Sheet
Series Escher's Knot
Expansions:
1
(1 - x)n+1
ln
1
1 - x
=

i=0
(Hn+i - Hn)
n + i
i
xi
,
1
x
-n
=

i=0
i
n
xi
,
xn
=

i=0
n
i
xi
, (ex
- 1)n
=

i=0
i
n
n!xi
i!
,
ln
1
1 - x
n
=

i=0
i
n
n!xi
i!
, x cot x =

i=0
(-4)i
B2ix2i
(2i)!
,
tan x =

i=1
(-1)i-1 22i
(22i
- 1)B2ix2i-1
(2i)!
, (x) =

i=1
1
ix
,
1
(x)
=

i=1
(i)
ix
,
(x - 1)
(x)
=

i=1
(i)
ix
,
Stieltjes Integration(x) =
p
1
1 - p-x
,
2
(x) =

i=1
d(i)
xi
where d(n) = d|n 1,
(x)(x - 1) =

i=1
S(i)
xi
where S(n) = d|n d,
(2n) =
22n-1
|B2n|
(2n)!
2n
, n  N,
x
sin x
=

i=0
(-1)i-1 (4i
- 2)B2ix2i
(2i)!
,
1 -

1 - 4x
2x
n
=

i=0
n(2i + n - 1)!
i!(n + i)!
xi
,
ex
sin x =

i=1
2i/2
sin i
4
i!
xi
,
1 -

1 - x
x
=

i=0
(4i)!
16i

2(2i)!(2i + 1)!
xi
,
arcsin x
x
2
=

i=0
4i
i!2
(i + 1)(2i + 1)!
x2i
.
If G is continuous in the interval [a, b] and F is nondecreasing then
b
a
G(x) dF(x)
exists. If a  b  c then
c
a
G(x) dF(x) =
b
a
G(x) dF(x) +
c
b
G(x) dF(x).
If the integrals involved exist
b
a
G(x) + H(x) dF(x) =
b
a
G(x) dF(x) +
b
a
H(x) dF(x),
b
a
G(x) d F(x) + H(x) =
b
a
G(x) dF(x) +
b
a
G(x) dH(x),
b
a
c  G(x) dF(x) =
b
a
G(x) d c  F(x) = c
b
a
G(x) dF(x),
b
a
G(x) dF(x) = G(b)F(b) - G(a)F(a) -
b
a
F(x) dG(x).
If the integrals involved exist, and F possesses a derivative F at every
point in [a, b] then
b
a
G(x) dF(x) =
b
a
G(x)F (x) dx.
Cramer's Rule
00 47 18 76 29 93 85 34 61 52
86 11 57 28 70 39 94 45 02 63
95 80 22 67 38 71 49 56 13 04
59 96 81 33 07 48 72 60 24 15
73 69 90 82 44 17 58 01 35 26
68 74 09 91 83 55 27 12 46 30
37 08 75 19 92 84 66 23 50 41
14 25 36 40 51 62 03 77 88 99
21 32 43 54 65 06 10 89 97 78
42 53 64 05 16 20 31 98 79 87
Fibonacci Numbers
If we have equations:
a1,1x1 + a1,2x2 +    + a1,nxn = b1
a2,1x1 + a2,2x2 +    + a2,nxn = b2
...
...
...
an,1x1 + an,2x2 +    + an,nxn = bn
Let A = (ai,j) and B be the column matrix (bi). Then
there is a unique solution iff det A = 0. Let Ai be A
with column i replaced by B. Then
xi =
det Ai
det A
.
1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, . . .
Definitions:
Fi = Fi-1+Fi-2, F0 = F1 = 1,
F-i = (-1)i-1
Fi,
Fi = 1
5
i
- ^i
,
Cassini's identity: for i > 0:
Fi+1Fi-1 - F2
i = (-1)i
.
Additive rule:
Fn+k = FkFn+1 + Fk-1Fn,
F2n = FnFn+1 + Fn-1Fn.
Calculation by matrices:
Fn-2 Fn-1
Fn-1 Fn
=
0 1
1 1
n
.
The Fibonacci number system:
Every integer n has a unique
representation
n = Fk1
+ Fk2
+    + Fkm
,
where ki  ki+1 + 2 for all i,
1  i < m and km  2.
Improvement makes strait roads, but the crooked
roads without Improvement, are roads of Genius.
­ William Blake (The Marriage of Heaven and Hell)