Complex Analysis/Residuals

Definition

Let $G\subseteq ℂ$ be a domain, $z_0\in G$, and $f$ a function that is holomorphic except for isolated singularities $S\subset G$, i.e., $f:G\setminus S\to ℂ$ is holomorphic. If $z_0\in S\subset G$ is an Isolated singularity of $f$ with $D_r(z_0)\cap S=z_0$, the residue is defined as:

$\mathrm{r}\mathrm{e}\mathrm{s}{z}_0(f):=\frac{1}{2\pi i}\int \partial D_r(z_0)f(\xi ),d\xi =\frac{1}{2\pi i}\underset{|\xi -z_0|=r}{\int }f(\xi ),d\xi$.

Relation between Residue and Laurent Series

If $f$ is expressed as a Laurent series around an isolated singularity $z_0\in S\subset G$, the residue can be computed as follows: With $f(z)=\sum _{n=-\mathrm{\infty }}^{\mathrm{\infty }}{a}_n(z-z_0{)}^n$ as the Laurent Expansion of $f$ around $z_0$, it holds:

$\mathrm{r}\mathrm{e}\mathrm{s}{z}_0(f)=\frac{1}{2\pi i}\int \partial D_r(z_0)f(\xi ),d\xi =\frac{1}{2\pi i}\underset{\partial D_r(z_0)}{\int }{a}_{-1}\cdot (\xi -z_0{)}^{-1},d\xi =\frac{1}{2\pi i}{a}_{-1}\cdot \underbrace{\underset{\partial D_r(z_0)}{\int }(\xi -z_0{)}^{-1},d\xi }=2\pi i=a-1$.

Here, it is assumed that the closed disk $\overline{D_r(z_0)}$ contains only the singularity $z_0\in S$, i.e., $\overline{D_r(z_0)}\cap S=z_0$.

Thus, the residue $\mathrm{r}\mathrm{e}\mathrm{s}{z}_0(f)=a-1$ can be identified as the coefficient of $(z-z_0{)}^{-1}$ in the Laurent series of $f$ around $z_0$.

Terminology

The residue (from Latin residuere - to remain) is named so because, during integration along the path $\gamma (t):=z_0+r\cdot e^{it}$ with $t\in [0,2\pi ]$ around $z_0$, the following holds:

$\begin{array}{cc}\underset{|w-z_0|=r}{\int }f(w)dw& =\sum _{n=-\mathrm{\infty }}^{+\mathrm{\infty }}{a}_n\underset{|w-z_0|=r}{\int }(w-z_0{)}^{n}dw\\ & =2\pi i\cdot a_{-1}\end{array}$

The residue is, therefore, what "remains" after integration.

Calculation for Poles

If $z_0\in U$ is a pole of order $m$ of $f$, the Laurent Expansion of $f$ around $z_0$ has the form:

$f(z)=\sum _{k=-m}^{\mathrm{\infty }}{a}_k(z-z_0{)}^k$

with $a_{-m}\ne 0$.

Proof 1: Removing the principal part by multiplication

By multiplying with $(z-z_0{)}^m$, we obtain:

$g_m(z):=(z-z_0{)}^m\cdot f(z)=\sum _{k=0}^{\mathrm{\infty }}{a}_{k-m}(z-z_0{)}^k$

The residue $a_{-1}$ is now the coefficient of $(z-z_0{)}^{m-1}$ in the power series of $g_m(z)$.

Proof 2: Using (m-1)-fold differentiation

Through $m-1$-fold differentiation, the first $m-1$ terms in the series, from $n=0$ to $m-2$, vanish. The residue is then the coefficient of $(z-z_0{)}^0$, yielding:

$g_m^{(m-1)}(z)=\sum _{k=m-1}^{\mathrm{\infty }}\frac{k!}{(k-m+1)!}{a}_{k-m}(z-z_0{)}^{k-m+1}$.

Proof 3: Limit process to find the coefficient of $(z-z_0{)}^0$

By shifting the index, we obtain:

$g_m^{(m-1)}(z)=\sum _{k=-1}^{\mathrm{\infty }}\frac{m+k!}{(k+1)!}{a}_k(z-z_0{)}^{k+1}$

Taking the limit $z\to z_0$, all terms with $k\ge 0$ vanish, yielding:

$\underset{z\to z_0}{\lim }{g}_m^{(m-1)}(z)=\frac{(m-1)!}{0!}\cdot a_{-1}\cdot (z-z_0{)}^0=(m-1)!\cdot a_{-1}$.

Thus, the residue can be computed using the limit $z\to z_0$:

$\mathrm{r}\mathrm{e}\mathrm{s}{z}_0(f)=a-1=\frac{1}{(m-1)!}\cdot \underset{z\to z_0}{\lim }{g}_m^{(m-1)}(z)$.

Tasks for Students

• Explain why, during integration of the Laurent series, all terms from the regular part and all terms with index $n\in \mathbb{Z}$ with $n\ne -1$ contribute

$\underset{\partial D_r(z_0)}{\int }{a}_n\cdot (\xi -z_0{)}^n,d\xi =0$.

• Why is it allowed to interchange the processes of integration and series expansion?

$\sum _{n=-\mathrm{\infty }}^{+\mathrm{\infty }}\underset{\partial D_r(z_0)}{\int }{a}_n(\xi -z_0{)}^{n}d\xi =\underset{\partial D_r(z_0)}{\int }\sum _{n=-\mathrm{\infty }}^{+\mathrm{\infty }}{a}_n(\xi -z_0{)}^{n}d\xi =\frac{1}{2\pi i}\underset{\partial D_r(z_0)}{\int }f(\xi ),d\xi ={\mathrm{r}\mathrm{e}\mathrm{s}}_{z_0}(f)$

• Given the function $f:ℂ\setminus i\to ℂ$ with $z\mapsto f(z)=\frac{e^{z-i}}{(z-i{)}^5}$, compute the residue ${\mathrm{r}\mathrm{e}\mathrm{s}}_{z_0}(f)$ with $z_0:=i$!.

See Also

• Residuals
• Laurent series examples

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• Source: Kurs:Funktionentheorie/Residuum - URL:

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• Date: 12/30/2024

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