Complex Analysis Part 2

Complex Integration to the end (wk 7-12)

C’ Path

gamma is a C’ Path if it is differentiable and its derivative is continuous

Smooth path

A path is smooth if it is infinitely differentiable

Integral over a C’ Path

∫fdz := ∫f(δ(t))*δ’(t)dt

(δ is the path, take the integral from the endpoints)

∫f(z) dz over p1+p2

∫f(z)dz over p1 + ∫g(z)dz over p2

Arc length of a path

l(p) = ∫|p’(t)|dt

Bounding integral lemma

| ∫f(z)dz | ≤ M * l(p), where M is a real number M>0 s.t. |f(z)| ≤ M
(M is a maximum of f(z))

Fundamental theorem of contour integrals

∫f(z) dz over p = F(p(b)) - F(p(a))

given: p is c’ path, f is continuous and defined on the same set as p, f has an integral.

T/F the integral of F over p is path dependent

F - depends only on endpoints

Morera Thoerem

∫f dz over p = 0,

given: p is a simple closed path, f is analytic.

∮ vs ∫

∮ is for closed paths.

Cauchy-Goursat Theorem

∮f(z)dz over C = 0.
where f is analytic in and on its boundary C.

Consequences of Cauchy’s Theorem

1. ∫ f(z)dz over p1 = ∫ f(z)dz over p2 (path independence)

2. G(z) = ∫f(z)dz, then G is analytic

Cauchy Integral Fomulas

1/2ipi ∫ f(z)/z-a dz = f(a)
and
the nth derivative of f at a = n!/2ipi ∫f(z)/(z-a)^(n+1) dz (provided the nth deriative exists)

gamma must be a closed curve, f must be holomorphic on and inside gamma

Cauchy’s Inequalities

| f^n (a) | = M * n!/p^n for n∈W
where M = max{ |f(z)| : |z-a| = r } (the maximum value of f at the border of the circle

Louiville’s Thoerem

Every bounded, entire function is constant.

Fundamental theorem of Algebra

every non-constant polynomial in C has at least one root in C

Gauss’ mean value theorem

f(a) = 1/2pi * ∫_p f(a+re^iθ) dθ
f os analytic in and on the circle |z-a| = r

Maximum Modulus Theorem

if f is analytic in and in a simple closed curve, f_max is on the boundary

Types of singularities

1. Isolated

2. Removable

3. Poles

4. Essential

Isolated Singularity

there are no other singularities right next to it

Removable Singularity

an isolated singularity where the lim f(z) at the singularity exists. check either by l’Hopital or factoring and cancelling.

Pole

isolated singularity where lim z→a of (z-a)ⁿ *f(z) exists and is non-zero

Essential Singularity

isolated singularity which is not removable or a pole

Argument Theorem

1/2ipi * ∮p f’/f dz = N - P
f is holomorphic and non-vanishing in and on p. N,P are counted with multiplicity

Rouche’s Theorem

if |g(z)| < |f(z)| ∀ z∈δ, then f and f+g have the same number of zeros inside δ

Entire/Analytic

differentiable at all points on the complex plane

Meromorphic

analytic everywhere except at a finite number of poles

Convergence Tests

1. Ratio

2. Root

3. Weierstass M Test

Ratio Test

let L:= lim | u_n+1 | / | u_n |, n→inf

if L<1, convergence

if L > 1 divergence

if L = 1, inconclusive

Root test

let L:= lim |u_n|^1/n

the sum of u_n :

converges if L < 1

diverges if L > 1

inconclusive if L = 1

Wierstrass M-Test

if |u_n| ≤ M_n and ∑₁^infM_n converges, then ∑₁ⁿ uniformly converges

Properties of uniformly convergent series

1. if u_n is holomorphic and ∑₁^infu_n is uniformly convergent, then ∑₁^infu_n is holomorphic

2. If u_n is holomorphic, then u’(z) = ∑₁^infu’_n (differentiate termwise)

3. Integrate termwise if u_n is continuous

Taylors Theorem

let f be holomorphic on and inside δ with center at w=a, then

f(w) = f(a) + f’(a) + f”(a)/2! (w-a)² + … + f⁽ⁿ⁾(a)/n!*(w-a)ⁿ

Laurent’s Theorem

f(z) = ∑₁^infa_-n/(z-a)ⁿ + ∑₀^infa_n(z-a)ⁿ = ∑_-inf^infa_n(z-a)ⁿ

Residue

f is holomorphic on and inside a circle except at the center a, then
f(z) = ∑_-inf^infa_n(z-a)ⁿ

and a_n = 1/2ipi * ∮ f(z) / (z-a)ⁿ⁺¹ dz,

a_n is called the residue.

Cauchy’s Residue Theorem

1/2ipi * ∮f(z)dz = sum of the residues

Residue Limit Formula

a_-1 = lim_z→a 1/(k-1)! d^k-1/dz^k-1 (z-a)^k * f(z)