Complex Analysis Bookwork

Question 1

What is a domain

Answer 1

A connected open subset of C

Question 2

Alternate differentiable criteria

Answer 2

differentiable at a iff

f(z)=f(a)+f’(a)(z-a)+ε(z)(z-a)

Question 3

Holomorphic def

Answer 3

f is holomorphic if it is complex differentiable at every point in it’s domain

Question 4

Cauchy Riemann (and proof outline)

Answer 4

dx(u)=dy(v); dy(u)=-dx(v)
f’(a)=dx(u)+idx(v)

Use alternate criterion and split into real and imaginary parts. For dx(u) then plug in y=y0 to get result

Question 5

Prove f’ identically 0 => f constant

Answer 5

Use Cauchy Riemann

Question 6

Prove components of holomorphic function are harmonic

Answer 6

Cauchy riemann

Question 7

Formula for radius of convergence

Answer 7

1/R = lim sup |an|^(1/n)

Question 8

Multi valued function def

Answer 8

A map f:U->P(C) assigning each point to a subset of the complex numbers

Question 9

Define a branch of a multifunction

Answer 9

A branch of a multifunction F on a subset V is a function g:V->C such that g(z) in f(z)

Question 10

Define a path

Answer 10

A continuous function from [a,b] to C

Question 11

What does it mean for 2 paths to be equivalent

Answer 11

There exists an S (with relevant domain & codomain) st P1=P2 composed S

Question 12

Prove triangle inequality for complex integrals. I.E.

|int[b][a] F dt|<=int[a][b]|F|dt

Answer 12

F is continuous so |F| integrable 
Let int[a][b] F dt = re^iθ
Then taking components in the e^iθ and ie^ iθ directions we have 
F=u(t)e^ iθ + iv(t)e^ iθ
Then by comparing we see
int[a][b] F dt = e^ iθ int[a][b] u dt
The use triangle inequality on u and
 |u|<=|F|

Question 13

Define the length of a path P:[a,b]->C

Answer 13

int[a][b] |P’(t)| dt

Question 14

Prove uniform convergence => integral convergence

Answer 14

|int[P] f dz - int[P] fn dz| = |int[P] f-fn dz | <= sup{|f(z)-fn(z)|}l(P) ->0 as fn->f uniformly

Question 15

Give an explicit formula for the stereographic projection

Answer 15

S(z) = (2x, 2y, x^2+y^2-1) / (x^2+y^2+1)

Question 16

Give an explicit expression for the metric on C that makes the stereographic projection an isometry

Answer 16

d(z,w)=
2|z-w|
(1+|z|^2)^0.5 * (1+|w|^2)^0.5

Question 17

Give the distance on C∞ from z to ∞

Answer 17

d(z,∞) =
2
root( 1+|z|^2 )

Question 18

Define a mobius map

Answer 18

The map Ψ(z)=az+b/cz+d
Where the matrix (a,b;c,d) is in GL2(C).
Also if c=0 then Ψ(∞)=∞ and if c not 0 then Ψ(-d/c)=0 and Ψ(∞)=a/c

Question 19

Prove every mobius map can be written as a series of translations, inversions and dilations

Answer 19

Page 9

Question 20

Define a harmonic conjugate of a harmonic function u

Answer 20

If f=u+iv is holomorphic then v is a harmonic conjugate of u

Question 21

Show the length of a path is invariant under reparametrization

Answer 21

Use chain rule basically

Question 22

Prove the fundamental theorem of calculus for complex path integrals

Answer 22

Loosely,
Integral of f’ over γ dz =
integral of f’(γ)γ’(t) dt =
integral of d/dt ( f(γ) ) dt =
fγ(1)-fγ(0)

Question 23

Prove the estimation Lena for complex path integrals

Answer 23

Note [a,b] compact so γ([a,b]) compact as γ continuous. Thus f is bounded on this set and sup|f| exists
Then take the integral into its dt form and use the result for reals