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Math 416 Final Review

Story after Cauchy's theorem

Chapter 7: Cauchy's Theorem

Existence of harmonic conjugate

Suppose f=u+iv is holomorphic on a domain U\subset\mathbb{C}. Then u=\Re f is harmonic on U. That is \Delta u=\frac{\partial^2 u}{\partial x^2}+\frac{\partial^2 u}{\partial y^2}=0.

Moreover, there exists g\in O(U) such that g is unique up to an additive imaginary constant.

Example:

Find a harmonic conjugate of u(x,y)=\log|\frac{z}{z-1}|

Note that \log(\frac{z}{z-1})=\log \left|\frac{z}{z-1}\right|+i(\arg(z)-\arg(z-1)) is harmonic on \mathbb{C}\setminus\{1\}.

So the harmonic conjugate of u(x,y)=\log|\frac{z}{z-1}| is v(x,y)=\arg(z)-\arg(z-1)+C where C is a constant.

Note that the harmonic conjugate may exist locally but not globally. (e.g. u(x,y)=\log|z(z-1)| has a local harmonic conjugate i(\arg(z)+\arg(z-1)+C) but this is not a well defined function since \arg(z)+\arg(z-1) is not single-valued.)

Corollary for harmonic functions

Theorem 7.19

Harmonic function are infinitely differentiable.

Theorem 7.20

Mean value property of harmonic functions.

Let u be harmonic on an open set of \Omega, then

u(z_0)=\frac{1}{2\pi}\int_0^{2\pi} u(z_0+re^{i\theta}) d\theta

for any z_0\in\Omega and r>0 such that D(z_0,r)\subset\Omega.

Theorem 7.21

Identity theorem for harmonic functions.

Let u be harmonic on a domain \Omega. If u=0 on some open set G\subset\Omega, then u\equiv 0 on \Omega.

Theorem 7.22

Maximum principle for harmonic functions.

Let u be a non-constant real-valued harmonic function on a domain \Omega. Then |u| does not attain a maximum value in \Omega.

Chapter 8: Laurent Series and Isolated Singularities

Laurent Series

Laurent series is a generalization of Taylor series.

Laurent series is a power series of the form

f(z)=\sum_{n=-\infty}^{\infty} a_n (z-z_0)^n

where

a_k=\frac{1}{2\pi i}\int_{C_r}\frac{f(z)}{(z-z_0)^{k+1}}dz

The series converges on an annulus R_1<|z-z_0|<R_2.

where R_1=\limsup_{n\to\infty} |a_{-n}|^{1/n} and R_2=\frac{1}{\limsup_{n\to\infty} |a_n|^{1/n}}.

Cauchy's Formula for an Annulus

Let f be holomorphic on an annulus R_1<r_1<|z-z_0|<r_2<R_2. And w\in A(z_0;R_1,R_2). Find the Annulus w\in A(z_0;r_1,r_2)

Then

f(w)=\frac{1}{2\pi i}\int_{C_{r_1}}\frac{f(z)}{z-w}dz-\frac{1}{2\pi i}\int_{C_{r_2}}\frac{f(z)}{z-w}dz

Isolated singularities

Let f be holomorphic on 0<|z-z_0|<R (The punctured disk of radius R centered at z_0). If there exists a Laurent series of f convergent on 0<|z-z_0|<R, then z_0 is called an isolated singularity of f.

The series f(z)=\sum_{n=-\infty}^{-1}a_n(z-z_0)^n is called the principle part of Laurent series of f at z_0.

Removable singularities

If the principle part of Laurent series of f at z_0 is zero, then z_0 is called a removable singularity of f.

Example:

f(z)=\frac{e^z-1}{z^2} has a removable singularity at z=0.

The Laurent series of f at z=0 can be found using the Taylor series of e^z-1 at z=0.

e^z-1=z+\frac{z^2}{2!}+\frac{z^3}{3!}+\cdots

So the Laurent series of f at z=0 is

f(z)=\frac{1}{z^2}+\frac{1}{z}+\sum_{n=0}^{\infty}\frac{z^n}{n!}

The principle part is zero, so z=0 is a removable singularity.

Poles