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De nitions Series
f n O g n i p ositive c n such that
f n cg n n n
X^ n
i
i
nn
X^ n
i
i^
nn n
X^ n
i
i^
n^ n
In general
X^ n
i
im^
m
n m^
Xn
i
i m^ im^ m im^
nX
i
im^
m
X^ m
k
m k
Bk nm k^
Geometric series
X^ n
i
ci^
cn^
c
c
X^
i
ci^
c
X^
i
ci^
c
c
c
X^ n
i
ici^
ncn^ n cn^ c
c ^
c
X^
i
ici^
c
c^
c
Harmonic series
Hn
X^ n
i
i
X^ n
i
iHi
nn
Hn
nn
X^ n
i
Hi n Hn n
X^ n
i
i m
Hi
n m
Hn
m
f n g n i p ositive c n such that
f n cg n n n
f n g n i f n O g n and f n g n
f n og n i limn f ng n
lim
n
an a i R n such that
jan aj n n
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 fai j i n i Ng
lim sup
n
an lim
n
sup fai j i n i Ng
n
k
Combinations Size k sub sets of a size n set
n
k
Stirling numb ers st kind Arrangements of an n ele ment set into k cycles
n k
n
n k k
X^ n
k
n k
n^
n k
n
n k
n k
n k
n
k
n k
n
k
n
k
n m
m k
n k
n k
m k
X
k n
r k k
r n n
X^ n
k
k m
n m
X^ n
k
r k
s
n k
r s n
n k
k
k n
k
n
n n
n
n ^
n k
k
n
k
n
k
n k Stirling^ numb^ ers^ ^ nd^ kind Partitions of an n element set into k nonempty sets n k st^ order^ Eulerian^ numb^ ers Permutations n on
f ng with k ascents
n k nd^ order^ Eulerian^ numb^ ers Cn Catlan Numb ers Binary trees with n vertices
n
n
n
n Hn
n n
n k
n k
n k
n
n
k
n
k
n
n
n
n
n
X^ n
k
n k
n Cn
n
n n
n
n
n
n k
n
n k
n k
k
n
k
n k
n
k
k
n
if k otherwise
n
n^ n
n
n^ n n^
n
xn^
X^ n
k
n k
x k n
n m
X^ m
k
n k
m k n^ k^ m
n m
X^ n
k
n k
k
n m
n m
X^ n
k
n k
n k
m
n k^ m^ k
n
n n
for n
n k
k
n
k
n k
n
k
X^ n
k
n k
nn n
x
x n
X^ n
k
n k
x n k
n
n m
X
k
n k
k m
X^ n
k
k m
m n k^
Identities Cont Trees
n m
X
k
n k
k m
X^ n
k
k m
nn k^ ^ n
X^ n
k
k
k m
x
x n
X^ n
k
n k
x k n
n m
X
k
n k
k m
n k^
n m
X
k
n k
k m
m k^
m n m
X^ m
k
k
n k k
m n m
X^ m
k
k n k
n k k
n m
X
k
n k
k m
m k^ n m
n m
X
k
n k
k m
m k^ for n m
n
n m
X
k
m n
m k
m n n k
m k k
n
n m
X
k
m n
m k
m n n k
m k k
n m
m
X
k
k
n k
m
n k
n m
m
X
k
k
n k
m
n k
Every tree with n
vertices has n
edges Kraft inequal ity If the depths of the leaves of a binary tree are d dn
X^ n
i
di
and equality holds only if every in ternal no de has sons
Recurrences Master metho d
T n aT nb f n a b
If such that f n O nlogb^ a ^
then T n nlogb^ a^
If f n nlogb^ a^ then T n nlog^ b^ a^ log n
If such that f n nlogb^ a ^
and c such that af nb cf n
for large n then T n f n
Substitution example Consider the following recurrence Ti
i
T i T
Note that Ti is always a p ower of two Let ti log Ti Then we have ti i^ ti t
Let ui ti i^ Dividing b oth sides of the previous equation by i^ we get ti i
i i ^
ti i Substituting we nd ui ui u
which is simply ui i So we nd that Ti has the closed form Ti i
i Summing factors example Consider the following recurrence Ti Tn n T n
Rewrite so that all terms involving T are on the left side
Ti Tn n
Now expand the recurrence and cho ose a factor which makes the left side tele scop e
T n T n n
T n T n n
log^ n ^
T T
log^ n^
T
Summing the left side we get T n Sum ming the right side we get
log X n
i
n i
i (^)
Let c and m log n Then we have
n
X^ m
i
ci^ n
cm^
c
nc clog^ n^
nc ck^ logc^ n^
nk^ ^ n n^ n
where k log ^ Full history recur
rences can often b e changed to limited his tory ones example Consider the follow ing recurrence
Ti
X^ i
j
Tj T
Note that
Ti
X^ i
j
Tj
Subtracting we nd
Ti Ti
X^ i
j
Tj
Xi
j
Tj
Ti
And so Ti Ti i^
Generating functions Multiply b oth sides of the equa tion by xi^ Sum b oth sides over all i for which the equation is valid Cho ose a generating function Gx Usually Gx
P
i x
i (^) Rewrite the equation in terms of the generating function Gx Solve for Gx The co ecient of xi^ in Gx is gi Example gi gi g
MultiplyX and sum
i
gi xi^
X
i
gi xi^
X
i
xi^
We cho ose Gx
P
i x
i (^) Rewrite in terms of Gx
Gx g
x
Gx
X
i
xi^
Simplify Gx x
Gx
x
Solve for Gx Gx x
x x
Expand this using partial fractions Gx x
x
x
x
X
i
i xi
X
i
xi
A
X
i
i^ xi^
So gi i^
Trigonometry Matrices More Trig
A
c B
a
b C
cos sin
Pythagorean theorem C ^ A^ B ^
De nitions sin a AC 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 inscrib ed circle AB^ ^
AB
A B C
Identities
sin x
csc x
cos x
sec x
tan x
cot x sin^ x cos ^ x
tan^ x sec ^ x cot^ x csc ^ x
sin x cos
^ x
sin x sin x
cos x cos x tan x cot
^ x
cot x cot x csc x cot x cot x
sinx y sin x cos y cos x sin y
cosx y cos x cos y sin x sin y
tanx y
tan x tan y tan x tan y
cotx y
cot x cot y cot x cot y
sin x sin x cos x sin x
tan x tan^ x
cos x cos ^ x sin^ x cos x cos^ x
cos x sin^ x cos x
tan^ x
tan^ x
tan x
tan x
tan^ x
cot x
cot^ x
cot x
sinx y sin x y sin^ x sin^ y
cosx y cos x y cos ^ x sin^ y
Eulers equation
eix^ cos x i sin x ei^
Multiplication
C A B cij
X^ n
k
aik bk j
Determinants det A i A is nonsingular
det A B det A det B
det A
X
Yn
i
sign ai i
and determinan t
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 bf g cdh
ceg f ha ibd
Permanents
p erm A
X
Yn
i
ai i
A
a
c
b
B
C
h
Law of cosines
c^ a^ b^ ab cos C
Area
A hc ab sin C
c^ sin A sin B sin C
Herons formula
A
p
s sa sb sc
s a b c
sa s a
sb s b
sc s c
More identities
sin x
r
cos x
cos x
r
cos x
tan x
r
cos x
cos x
cos x
sin x
sin x cos x
cot x
r
cos x
cos x
cos x sin x
sin x
cos x
sin x
eix^ e ix
i
cos x
eix^ e ix
tan x i
eix^ e ix
eix^ e ix^
i
eix^
eix^
sin x
sinh ix i
cos x cosh ix
tan x tanh ix i
Hyp erb olic Functions De nitions
sinh x
ex^ e x
cosh x
ex^ e x
tanh x
ex^ e x
ex^ e x^
csch x
sinh x
sech x
cosh x
coth x
tanh x
Identities
cosh ^ x sinh^ x tanh^ x sech ^ x
coth ^ x csch ^ x sinhx sinh x
cosh x cosh x tanhx tanh x
sinhx y sinh x cosh y cosh x sinh y
cosh x y cosh x cosh y sinh x sinh y
sinh x sinh x cosh x
cosh x cosh^ x sinh^ x
cosh x sinh x ex^ cosh x sinh x e x^
cosh x sinh xn^ cosh nx sinh nx n Z
sinh^ x cosh x cosh^ x cosh x
sin cos tan
p
p
p
p
p
p
^ ^
in mathematics you dont under stand things you just get used to them
c by Steve Seiden ^ J^ von^ Neumann
sseiden icsuciedu httpwwwicsuciedusseiden
Numb er Theory Graph Theory
The Chinese remainder theorem There ex ists a numb er C such that
C r mo d m
C rn mo d mn
if mi and mj are relatively prime for i j
Eulers function x is the numb er of p ositive integers less than x relatively prime to x If
Q n
i p
ei i is^ the^ prime^ fac torization of x then
x
Y^ n
i
pe i i^ pi
Eulers theorem If a and b are relatively prime then
a b^ mo d b
Fermats theorem
ap ^ mo d p
The Euclidean algorithm if a b are in tegers then gcd a b gcd a mo d b b
If
Q n
i p
ei i is^ the^ prime^ factorization^ of^ x then
S x
X
djx
d
Y^ n
i
pe i i^
pi
Perfect Numb ers x is an even p erfect num
b er i x n ^ n^ and n^ is prime
Wilsons theorem n is a prime i
n mo d n
M obius inversion
i
if i if i is not squarefree
r^ if i is the pro duct of
r distinct primes
If Ga
X
dja
F d
then F a
X
dja
dG
a
d
Prime numb ers
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^
n ln n
O
n ln n
De nitions Loop An edge connecting a ver tex to itself Directed Each edge has a direction Simple Graph with no lo ops or multiedges Walk A sequence v e v e v Trail A walk with distinct edges Path A trail with distinct vertices Connected A graph where there exists a path b etween any two vertices Component A maximal connected subgraph Tree A connected acyclic graph Free tree A tree with no ro ot DAG Directed acyclic graph Eulerian Graph with a trail visiting each edge exactly once Hamiltonian Graph with a path visiting each vertex exactly once Cut A set of edges whose re moval increases the num b er of comp onents Cut set A minimal cut Cut edge A size cut k Connected A graph connected with
the removal of any k
vertices
k Tough S V S we have
k cG S jS j
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 b e em b eded in the plane Plane graph An emb edding of a planar graph
X
v V
deg v m
If G is planar then n m f so
f n m n
Any planar graph has a vertex with de
gree
Notation E G Edge set V G Vertex set cG Numb er of comp onents G S Induced subgraph deg v Degree of v G Maximum degree G Minimum degree G Chromatic numb er E G Edge chromatic numb er Gc^ Complement graph Kn Complete graph Kn n Complete bipartite graph rk Ramsey numb er
Geometry Pro jective co ordinates triples x y z not all x y and z zero
x y z cx cy cz c
Cartesian Pro jective x y x y
y mx b m b
x c c
Distance formula Lp and L
metric
p
x x ^ x x ^
jx x jp^ jx x jp^
p
lim
p
jx x jp^ jx x jp^
p
Area of triangle x y x y and x y abs
x x y y
x x y y
Angle formed by three p oints
x y
x y
^
cos
x y x y
Line through two p oints x y and x y
x y x y x y
Area of circle volume of sphere A r ^ V r ^ If I have seen farther than others it is b ecause I have sto o d on the shoulders of giants Issac Newton
Calculus Cont
Z
arccos x a dx arccos x a
p
a^ x^ a
Z
arctan x a dx x arctan x a a lna^ x^ a
Z
sin^ axdx (^) a
ax sinax cos ax
Z
cos ^ axdx (^) a
ax sinax cosax
Z
sec ^ x dx tan x
Z
csc ^ x dx cot x
Z
sinn^ x dx
sinn ^ x cos x
n
n
n
Z
sinn ^ x dx
Z
cosn^ x dx
cos n ^ x sin x
n
n
n
Z
cosn ^ x dx
Z
tann^ x dx
tann ^ x
n
Z
tann ^ x dx n
Z
cotn^ x dx
cot n ^ x
n
Z
cotn ^ x dx n
Z
sec n^ x dx
tan x sec n ^ x
n
n
n
Z
sec n ^ x dx n
Z
csc n^ x dx
cot x csc n ^ x
n
n
n
Z
csc n ^ x dx n
Z
sinh x dx cosh x
Z
cosh x dx sinh x
Z
tanh x dx ln j cosh xj
Z
coth x dx ln j sinh xj
Z
sech x dx arctan sinh x
Z
csch x dx ln
tanh x
Z
sinh^ x dx sinh x x
Z
cosh^ x dx sinh x x
Z
sech ^ x dx tanh x
Z
arcsinh x a dx x arcsinh x a
p
x^ a^ a
Z
arctanh x a dx x arctanh x a a ln ja^ x^ j
Z
arccosh x a dx
x arccosh
x a
p
x^ a^ if arccosh x a and a
x arccosh
x a
p
x^ a^ if arccosh x a and a
Z
dx
p
a^ x^
ln
x
p
a^ x
a
Z
dx a^ x^
a arctan x a a
Z p
a^ x^ dx x
p
a^ x^ a
arcsin^
x a ^ a^ ^
Z
a^ x^ ^ dx x a^ x^
p
a^ x^ a
arcsin^
x a ^ a^ ^
Z
dx
p
a^ x^
arcsin x a a
Z
dx
a^ x^
a
ln
a x
a x
^
Z
dx
a^ x^ ^
x a^
p
a^ x^
Z p
a^ x^ dx x
p
a^ x^ a
ln
x
p
a^ x
Z
dx
p
x^ a^
ln
x
p
x^ a
a
Z
dx ax^ bx
a
ln
x a bx
^ ^
Z
x
p
a bx dx
bx aa bx
b^
Z p
a bx x
dx
p
a bx a
Z
x
p
a bx
dx
Z
x
p
a bx
dx
p ln
p
a bx
p
a
p
a bx
p
a
^ a^ ^
Z p
a^ x
x
dx
p
a^ x^ a ln
a
p
a^ x
x
Z
x
p
a^ x^ dx a^ x^ ^
Z
x^
p
a^ x^ dx x x^ a^
p
a^ x^ a
arcsin^
x a ^ a^ ^ ^
Z
dx
p
a^ x^
a ln
a
p
a^ x
x
Z
x dx
p
a^ x^
p
a^ x^
Z
x^ dx
p
a^ x^
x
p
a^ x^ a
arcsin^
x a a^ ^
Z p
a^ x x
dx
p
a^ x^ a ln
a
p
a^ x x
Z p
x^ a
x
dx
p
x^ a^ a arccos jaxj a
Z
x
p
x^ a^ dx x^ a^ ^
Z
dx x
p
x^ a^
a ln
x a
p
a^ x
Calculus Cont Finite Calculus
Z
dx x
p
x^ a^
a arccos jaxj a
Z
dx x^
p
x^ a^
p
x^ a a^ x
Z
x dx
p
x^ a^
p
x^ a^
Z p
x^ a x^
dx x^ a^ a^ x^
Z
dx ax^ bx c
p
b^ ac
ln
ax b
p
b^ ac
ax b
p
b^ ac
if b^ ac
p
ac b^
arctan
ax b
p
ac b^
if b^ ac
Z
dx
p
ax^ bx c
p
a
ln
ax^ ^ b^ ^
p
a
p
ax^ bx c
^ if^ a^ ^
p
a
arcsin
ax b
p
b^ ac
if a
Z p
ax^ bx c dx ax b a
p
ax^ bx c
ax b
a
Z
dx
p
ax^ bx c
Z
x dx
p
ax^ bx c
p
ax^ bx c a
b a
Z
dx
p
ax^ bx c
Z
dx x
p
ax^ bx c
p
c
ln
p
c
p
ax^ bx c bx c x
if c
p
c
arcsin
bx c
jxj
p
b^ ac
if c
Z
x^
p
x^ a^ dx x^ a^ x^ a^ ^
Z
xn^ sinax dx a xn^ cos ax n a
Z
xn ^ cos ax dx
Z
xn^ cos ax dx a xn^ sinax n a
Z
xn ^ sinax dx
Z
xn^ eax^ dx
xn^ eax a
n a
Z
xn ^ eax^ dx
Z
xn^ lnax dx xn
lnax n
n
Z
xn^ ln axm^ dx xn n
ln axm^
m n
Z
xn^ ln axm ^ dx
Dierence shift op erators
f x f x f x
E f^ x^ ^ f^ x^ ^ Fundamental Theorem
f x F x
X
f x x F x C
X^ b
a
f x x
bX
ia
f i
Dierences cu c u u v u v uv u v (^) E v u
xn^ nxn ^
Hx x ^ x^ x^
cx^ c cx^
x
m
x
m
Sums
P
cu x c
P
u x
P
u v x
P
u x
P
v x
P
u v x uv
P
E v^ u^ ^ x
P
xn^ x x
n m ^
P
x ^ x Hx
P
cx^ x c
x
c ^
P x
m
x
x
m
Falling Factorial Powers
xn^ xx x m n
x ^
xn^
x x jnj
n
xnm^ xm^ x mn^
Rising Factorial Powers
xn^ xx x m n
x ^
xn^
x x jnj
n
xnm^ xm^ x mn^ Conversion
xn^ n^ xn^ x m n
x n^
xn^ n^ xn^ x m n
x n^
xn^
X^ n
k
n k
xk^
X^ n
k
n k
n k^ xk^
xn^
X^ n
k
n k
n k^ xk^
xn^
X^ n
k
n k
xk^
x^ x^ x
x^ x^ x^ x^ x
x^ x^ x^ x^ x^ x^ x
x^ x^ x^ x^ x^ x^ x^ x^ x
x^ x^ x^ x^ x^ x^ x^ x^ x^ x^ x
x^ x^ x^ x
x^ x^ x^ x^ x^ x
x^ x^ x^ x^ x^ x^ x^ x
x^ x^ x^ x^ x^ x^ x^ x^ x^ x
x^ x^ x^ x^ x^ x^ x^ x^ x^ x^ x^ x
Series Eschers Knot
Expansions
xn^
ln
x
X^
i
Hni Hn
n i i
xi^
x
n
X^
i
i n
xi^
xn^
X^
i
n i
xi^ ex^ n^
X^
i
i n
nxi i
ln
x
n
X^
i
i n
nxi i
x cot x
X^
i
i^ Bi xi
i
tan x
X^
i
i ^
i i Bi xi
i
x
X^
i
ix^
x
X^
i
i ix^
x
x
X^
i
i ix^
x (^) Stieltjes Integration
Y
p
p x^
^ x
X^
i
di xi^
where dn
P
djn
x x
X^
i
S i xi^
where S n
P
djn d
n
n jBn j
n
n^ n N
x sin x
X^
i
i ^
i^ Bi xi
i
p
x
x
n
X^
i
n i n
in i
xi^
ex^ sin x
X^
i
i (^) sin i i
xi^
s
p
x
x
X^
i
i i^
p
i i
xi^
arcsin x x
X^
i
i (^) i i i xi^
If G is continuous in the interval a b and F is nondecreasing then
Z b
a
Gx dF x
exists If a b c then
Z c
a
Gx dF x
Z b
a
Gx dF x
Z c
b
Gx dF x
If the integrals involved exist
Z b
a
Gx H x
dF x
Z b
a
Gx dF x
Z b
a
H x dF x
Z b
a
Gx d
F x H x
Z b
a
Gx dF x
Z b
a
Gx dH x
Z b
a
c Gx dF x
Z b
a
Gx d
c F x
c
Z b
a
Gx dF x
Z b
a
Gx dF x GbF b GaF a
Z b
a
F x dGx
If the integrals involved exist and F p ossesses a derivative F ^ at every
p oint in a b then
Z b
a
Gx dF x
Z b
a
GxF ^ x dx
Crammers Rule
Fib onacci Numb ers If we have equations
a x a x an xn b
a x a x an xn b
an x an x ann xn bn
Let A aij and B b e the column matrix bi Then
there is a unique solution i det A Let Ai b e A
with column i replaced by B Then xi
det Ai det A
De nitions
Fi Fi Fi F F
F i i ^ Fi
Fi p
i i
Cassinis identity for i
Fi Fi F i i^
Additive rule
Fnk Fk Fn Fk Fn
Fn Fn Fn Fn Fn
Calculation by matrices
Fn Fn
Fn Fn
n
The Fib onacci numb er system Every integer n has a unique representation
n Fk Fk Fkm
where ki ki for all i
i m and km
Improvement makes strait roads but the cro oked roads without Improvement are roads of Genius William Blake The Marriage of Heaven and Hell
