Math416 Lecture 18

Chapter 8: Laurent Series and Isolated Singularities

8.1 Laurent Series

Definition of Laurent Series

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

where $c_n$ are complex coefficients.

Let $R_2=\frac{1}{\limsup_{n\to\infty} |c_n|^{1/n}}$ , then the Laurent series converges on $|z-z_0|<R_2$

Where $R_1=\limsup_{n\to-\infty} |c_n|^{-1/n}$ , if $|z-z_0|>R_1$ , the Laurent series diverges.

If $R_1\leq R_2$ , then the Laurent series converges on $A(z_0;R_1,R_2)=\{z:R_1<|z-z_0|<R_2\}$ , the Laurent series converges absolutely on $A(z_0;R_1,R_2)$

By Weierstrass, the limit is a holomorphic function on $A(z_0;R_1,R_2)$

If $R_1<r<R_2$ , then

\int_{C(z_0,r)} f(z) dz = \sum_{n=-\infty}^{\infty} c_n \int_{C(z_0,r)} (z-z_0)^n dz

Additional Proof

\int_{C(z_0,r)} (z-z_0)^n dz = \begin{cases} 2\pi i, & n=-1 \\0, & n\neq -1\end{cases}

Proof:

$\gamma(t)=z_0+re^{it}, t\in[0,2\pi]$

\begin{aligned} \int_{C(z_0,r)} (z-z_0)^n dz &= \int_0^{2\pi} (z_0+re^{it}-z_0)^n ire^{it} dt \\ &= ir^{n+1} \int_0^{2\pi} e^{i(n+1)t} dt \\ &= \begin{cases} 2\pi i, & n=-1 \\ \int_0^{2\pi} e^{i(n+1)t} dt = \frac{1}{i(n+1)}e^{i(n+1)t}\Big|_0^{2\pi} = 0, & n\neq -1 \end{cases} \end{aligned}

So,

\int_{C(z_0,r)} f(z) dz = \sum_{n=-\infty}^{\infty} c_n \int_{C(z_0,r)} (z-z_0)^n dz=c_{-1}2\pi i

And,

\int_{C(z_0,r)} f(z)(z-z_0)^k dz = \sum_{n=-\infty}^{\infty} c_n \int_{C(z_0,r)} (z-z_0)^{n+k} dz = c_{-1-k}2\pi i

So,

2\pi i c_j = \int_{C(z_0,r)} f(z)(z-z_0)^{-j-1} dz

Cauchy integral

Recall Cauchy integral formula:

f(z) = \int_{\gamma} \frac{\phi(\xi)}{\xi-z} d\xi

where $\gamma$ is a closed curve.

Suppose $|z-z_0|>R$ ,

\begin{aligned} \frac{1}{\xi-z}&=\frac{1}{\xi-z_0-(z-z_0)}\\ &=-\frac{1}{z-z_0}\frac{1}{1-\frac{\xi-z_0}{z-z_0}}\\ &=-\frac{1}{z-z_0}\sum_{n=0}^{\infty} \frac{(\xi-z_0)^n}{(z-z_0)^n}\\ &=-\sum_{m=1}^{\infty} (\xi-z_0)^{m-1}(z-z_0)^{-m} \end{aligned}

So,

\begin{aligned} f(z) &= \int_{\gamma} \frac{\phi(\xi)}{\xi-z} d\xi\\ &= -\int_{\gamma} \sum_{m=1}^{\infty} (\xi-z_0)^{m-1}(z-z_0)^{-m}\phi(\xi) d\xi\\ &=-\sum_{m=1}^{\infty} (z-z_0)^{-m} \int_{\gamma} (\xi-z_0)^{m-1} \phi(\xi) d\xi \end{aligned}

So the Cauchy integral $\int_{\gamma} \frac{\phi(\xi)}{\xi-z} d\xi$ is a convergent power series in $B_{d(z_0,\gamma)}(z_0)$ and is a convergent Laurent series (with just negative powers) in $\mathbb{C}\setminus B_{\max_{\xi\in \gamma} d(z_0,\xi)}(z_0)$

Theorem 8.4 Cauchy Theorem for Annulus

Suppose $f$ is holomorphic on $A(z_0;R_1,R_2)$ , Let $C_r=\{z:|z-z_0|=r\}$ , oriented counterclockwise. Then $I(r)=\int_{C_r} f(z) dz$ is independent of $r$ for $R_1<r<R_2$

Proof:

If integrand is continuous with respect to $r$ and continuous with respect to $t$ , then we can differentiate under the integral sign (Check after class, on Appendix 4?)

\begin{aligned} I(r)&=\int_0^{2\pi} f(z_0+re^{it})ire^{it}dt\\ \frac{dI}{dr}&=i\int_0^{2\pi}\frac{\partial}{\partial r}[f(z_0+re^{it})re^{it}]dt \end{aligned}

\frac{\partial}{\partial r}[f(z_0+re^{it})ire^{it}]=if'(z_0+re^{it})e^{it}+if(z_0+re^{it})e^{it}

\frac{\partial}{\partial t}[f(z_0+re^{it})e^{it}]=f'(z_0+re^{it})ire^{it}+f(z_0+re^{it})ire^{it}

This gives

\frac{\partial}{\partial r}[f(z_0+re^{it})ire^{it}]=\frac{\partial}{\partial t}[f(z_0+re^{it})e^{it}]

So,

\frac{dI}{dr}=i\int_0^{2\pi}\frac{\partial}{\partial r}[f(z_0+re^{it})ire^{it}]dt=i\int_0^{2\pi}\frac{\partial}{\partial t}[f(z_0+re^{it})e^{it}]dt=0

is a integration on a closed curve, so it is $0$ .

So, $I(r)$ is constant.

QED

Let $f$ be holomorphic on $A(z_0;R_1,R_2)$ . Let $C_r=\{z:|z-z_0|=r\}$ , oriented counterclockwise. Let $w\in A(z_0;R_1,R_2)$ . Choose $R_1<r_1<|w-z_0|<r_2<R_2$ such that $w\in A(z_0;r_1,r_2)$ . Then,

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

Proof:

Define $g(z)=\begin{cases} \frac{f(z)-f(w)}{z-w}, & z\neq w \\ f'(w), & z=w \end{cases}$

Then $g$ is holomorphic on $A(z_0;r_1,r_2)$ since $f$ is analytic at $w$ , $f(z)=\sum_{n=0}^{\infty} \frac{f^{(n)}(w)}{n!}(z-w)^n$

So,

f(z)-f(w)=f(w)+f'(w)(z-w)+\sum_{n=2}^{\infty} \frac{f^{(n)}(w)}{n!}(z-w)^n-f(w)=f'(w)(z-w)+\sum_{n=2}^{\infty} \frac{f^{(n)}(w)}{n!}(z-w)^n

So,

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

So,

\lim_{z\to w} \frac{f(z)-f(w)}{z-w}=f'(w)

So $\int_{C_{r_2}} g(z) dz=\int_{C_{r_1}} g(z) dz$ ,

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

Since $\int_{C_{r_2}} \frac{1}{z-w} dz=2\pi i$ and $\int_{C_{r_1}} \frac{1}{z-w} dz=0$ , (using Cauchy integral theorem on convex region)

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

QED

Since $\int_{C_{r_1}} \frac{f(z)}{z-w} dz$ is a Laurent series in negative powers which converges in $\mathbb{C}\setminus \overline{B_{r_1}(z_0)}$ , we can conclude that

$f(z)$ is given by a convergent Laurent series $\sum_{n=-\infty}^{\infty} a_n (z-z_0)^n$ in $\mathbb{C}\setminus \overline{B_{r_1}(z_0)}$ where $a_n=\frac{1}{2\pi i}\int_{C_r} \frac{f(z)}{(z-z_0)^{-1-n}} dz$

Laurent series converges in $A(z_0;R_1,R_2)$