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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.

Criterion for a removable singularity:

If f is bounded on 0<|z-z_0|<R, then z_0 is a removable singularity.

Example:

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

The Laurent series of f at z=0 is

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

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

Poles

If the principle part of Laurent series of f at z_0 is a finite sum, then z_0 is called a pole of f.

Criterion for a pole:

If f has an isolated singularity at z_0, and \lim_{z\to z_0}|f(z)|=\infty, then z_0 is a pole of f.

Example:

f(z)=\frac{1}{z^2} has a pole at z=0.

The Laurent series of f at z=0 is

f(z)=\frac{1}{z^2}

The principle part is \frac{1}{z^2}, so z=0 is a pole.

Essential singularities

If f has an isolated singularity at z_0, and it is neither a removable singularity nor a pole, then z_0 is called an essential singularity of f.

"Criterion" for an essential singularity:

If the principle part of Laurent series of f at z_0 has infinitely many non-zero coefficients corresponding to negative powers of z-z_0, then z_0 is called an essential singularity of f.

Example:

f(z)=\sin(\frac{1}{z}) has an essential singularity at z=0.

The Laurent series of f at z=0 is

f(z)=\frac{1}{z}-\frac{1}{6z^3}+\frac{1}{120z^5}-\cdots

Since there are infinitely many non-zero coefficients corresponding to negative powers of z, z=0 is an essential singularity.

Singularities at infinity

We say f has a singularity (removable, pole, or essential) at infinity if f(1/z) has an isolated singularity (removable, pole, or essential) at z=0.

Example:

f(z)=\frac{z^4}{(z-1)(z-3)} has a pole of order 2 at infinity.

Plug in z=1/w, we get f(1/w)=\frac{1}{w^2}\frac{1}{(1/w-1)(1/w-3)}=\frac{1}{w^2}\frac{1}{(1-w)(1-3w)}=\frac{1}{w^2}(1+O(w)), which has pole of order 2 at w=0.

Residue

The residue of f at z_0 is the coefficient of the term (z-z_0)^{-1} in the Laurent series of f at z_0.

Example:

f(z)=\frac{1}{z^2} has a residue of 0 at z=0.

f(z)=\frac{z^3}{z-1} has a residue of 1 at z=1.

Residue for pole with higher order:

If f has a pole of order n at z_0, then the residue of f at z_0 is

\operatorname{res}(f,z_0)=\frac{1}{(n-1)!}\lim_{z\to z_0}\frac{d^{n-1}}{dz^{n-1}}((z-z_0)^nf(z))

Chapter 9: Generalized Cauchy's Theorem

Winding number

The winding number of a closed curve C with respect to a point z_0 is

\operatorname{ind}_C(z_0)=\frac{1}{2\pi i}\int_C\frac{1}{z-z_0}dz

the winding number is the number of times the curve C winds around the point z_0 counterclockwise. (May be negative)

Contour integrals

A contour is a piecewise continuous curve \gamma:[a,b]\to\mathbb{C} with integer coefficients.

\Gamma=\sum_{i=1}^p n_j\gamma_j

where \gamma_j:[a_j,b_j]\to\mathbb{C} is continuous closed curve and n_j\in\mathbb{Z}.

Interior of a curve

The interior of a curve is the set of points z\in\mathbb{C} such that \operatorname{ind}_{\Gamma}(z)\neq 0.

The winding number of contour \Gamma is the sum of the winding numbers of the components of \Gamma around z_0.

\operatorname{ind}_{\Gamma}(z)=\sum_{j=1}^p n_j\operatorname{ind}_{\gamma_j}(z)

Separation lemma

Let \Omega\subseteq\mathbb{C} be a domain and K\subset \Omega be compact. Then there exists a simple contour \Gamma\subset \Omega\setminus K such that K\subset \operatorname{int}_{\Gamma}(\Gamma)\subset \Omega.

Key idea:

Let 0<\delta<d(K,\partial \Omega), then draw the grid lines and trace the contour.

Residue theorem

Let \Omega be a domain, \Gamma be a contour such that \Gamma\cap \operatorname{int}(\Gamma)\subset \Omega. Let f be holomorphic on \Omega\setminus \{z_1,z_2,\cdots,z_p\} and z_1,z_2,\cdots,z_p are finitely many points in \Omega, where z_1,z_2,\cdots,z_p\notin \Gamma. Then

\int_{\Gamma}f(z)dz=2\pi i\sum_{j=1}^p \operatorname{res}(f,z_j)

Key: Prove by circle around each singularity and connect them using two way paths.

Homotopy*

Suppose \gamma_0, \gamma_1 are two curves from [0,1] to \Omega with same end points P,Q.

A homotopy is a continuous function of curves \gamma_t, 0\leq t\leq 1, deforming \gamma_0 to \gamma_1, keeping the end points fixed.

Formally, if H:[0,1]\times [0,1]\to \Omega is a continuous function satsifying

1. H(s,0)=\gamma_0(s), \forall s\in [0,1]
2. H(s,1)=\gamma_1(s), \forall s\in [0,1]
3. H(0,t)=P, \forall t\in [0,1]
4. H(1,t)=Q, \forall t\in [0,1]

Then we say H is a homotopy between \gamma_0 and \gamma_1. (If \gamma_0 and \gamma_1 are closed curves, Q=P)

Lemma 9.12 Technical Lemma

Let \phi:[0,1]\times [0,1]\to \mathbb{C}\setminus \{0\} is continuous. Then there exists a continuous map \psi:[0,1]\times [0,1]\to \mathbb{C} such that e^\phi=\psi. Moreover, \psi is unique up to an additive constant in 2\pi i\mathbb{Z}.

General approach to evaluate definite integrals

Choose a contour so that one side is the desired integral.

Handle the other sides using:

• Symmetry
• Favorite estimate
• Bound function by another function whose integral goes to 0