Complex Analysis/Residue Theorem

The residue theorem states, how to calculate the integral of a holomorphic function using its Residuals .

Statement

Let $f$ be a holomorphic function in a region $G$ except for a discrete set of isolated singularities $S\subseteq G$, and let $\mathrm{\Gamma }$ be a null-homologous Chain in $G$ that does not intersect any point of $S$. Then, the following holds:

$\underset{\mathrm{\Gamma }}{\int }f(z)dz=2\pi i\cdot \sum _{z\in S}n(\mathrm{\Gamma },z)\cdot {\mathrm{r}\mathrm{e}\mathrm{s}}_z(f).$

Proof

The sum in the statement of the residue theorem is finite because $\mathrm{\Gamma }$ can enclose only finitely many points of the discrete set $S$ of singularities.

Step 1 - Reduction to a finite number of summands

Let $z_1,\dots ,z_k$ be the points in $S$ for which $n(\mathrm{\Gamma },z_i)\ne 0$. The singularities in $S$ that are not enclosed are denoted by $T:=z\in S:n(\mathrm{\Gamma },z)=0$.

Step 2 - Nullhomologous cycle

$\mathrm{\Gamma }$ is assumed to be null-homologous in $G$. By the definition of $T$,is $\mathrm{\Gamma }$ also null-homologous in $G\setminus T$.

Step 3 - Principal parts of the Laurent series

For the singularities $z_i\in S$ with $\mu (\mathrm{\Gamma },z_i)=0$ and $1\le i\le k$, let

$h_i(z)=\sum _{j=-1}^{-\mathrm{\infty }}{a}_{ij}(z-z_i{)}^j$

be the main part of the Laurent Expansion of $f$ around $z_i$. The function $h_i$ is holomorphic on $ℂ\setminus \{z_i\}$.

Step 4 - Subtraction of principal parts

Subtracting all the principal parts $h_i$ corresponding to $z_i$ from the given function $f$, we obtain

$g:=f-\sum _{i=1}^k{h}_i$

a function on $G\setminus T$ that now has only removable singularities.

Step 5 - Holomorphic extension to $G$

If the singularities $z_i$ are Isolated singularity on $G\setminus T$, $g$ can be extended holomorphically to all $z_1,\dots ,z_k\in G\setminus T$.

Step 6 - Application of Cauchy's integral theorem

By the Cauchy Integral Theorem for $\mathrm{\Gamma }$, we have

$\underset{\mathrm{\Gamma }}{\int }g(z)dz=0$

so, by the definition of $g$,

$\begin{array}{ccc}\underset{\mathrm{\Gamma }}{\int }f(z)dz& =& \underset{\mathrm{\Gamma }}{\int }g(z)dz+\sum _{i=1}^k\underset{\mathrm{\Gamma }}{\int }{h}_i(z)dz\\ & =& \sum _{i=1}^k\underset{\mathrm{\Gamma }}{\int }{h}_i(z)dz.\end{array}$

Step 7 - Calculation of integrals of the principal parts

The computation of the integral over $f$ reduces to computing the integrals of the principal parts $h_i$ for $1\le i\le k$. Using the linearity of the integral, we have:

$\begin{array}{ccc}\underset{\mathrm{\Gamma }}{\int }{h}_i(z)dz& =& \sum _{j=-\mathrm{\infty }}^{-i}{a}_{ij}\underset{\mathrm{\Gamma }}{\int }(z-z_i{)}^j\end{array}$

the terms $\underset{\mathrm{\Gamma }}{\int }(z-z_i{)}^j$ For $j\le -2$, have antiderivatives, so $\underset{\mathrm{\Gamma }}{\int }(z-z_i{)}^j=0$.

Step 8 - Calculation of integrals of the principal parts

Finally, the computation of the integrals of the principal parts yields, using the definition of the Winding number:

$\begin{array}{ccc}\underset{\mathrm{\Gamma }}{\int }{h}_i(z)dz& =& \sum _{j=-\mathrm{\infty }}^{-i}{a}_{ij}\underset{\mathrm{\Gamma }}{\int }(z-z_i{)}^j\\ & =& a_{i,-1}\underset{\mathrm{\Gamma }}{\int }(z-z_i{)}^{-1}dz\\ & =& a_{i,-1}\cdot 2\pi \cdot i\cdot n(\mathrm{\Gamma },z_i)\\ & =& 2\pi \cdot i\cdot n(\mathrm{\Gamma },z_i)\cdot {\mathrm{r}\mathrm{e}\mathrm{s}}_{z_i}f\end{array}$

after.

Step 9 - Calculation of the integrals of the residues

Thus, the statement follows as:

$\begin{array}{ccc}\underset{\mathrm{\Gamma }}{\int }f(z)dz& =& \sum _{i=1}^k\underset{\mathrm{\Gamma }}{\int }{h}_i(z)dz\\ & =& 2\pi i\sum _{i=1}^{k}n(\mathrm{\Gamma },z_i){\mathrm{r}\mathrm{e}\mathrm{s}}_{z_i}f\end{array}$

Questions about the residue theorem

• Let $f:G\to \widehat{ℂ}$ be a meromorphic function (i.e., holomorphic except for a discrete set of singularities in $G$). Why does the cycle $\mathrm{\Gamma }$ enclose only finitely many poles?

Applications

The Zeros and poles counting integral counts the zeros and poles of a meromorphic function.

See also

• null-homologous
• Isolated singularity

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• Date: 01/05/2024

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