Complex Analysis/Real integrals with residue theorem

Introduction

The residue theorem in complex analysis applies to null-homologous cycles in regions with isolated singularities. To use the residue theorem to calculate integrals, a real integral is extended to a null-homologous cycle in the complex plane, and the residue theorem is applied to it.

General Procedure

• First, the real integral is interpreted as a line integral on the real axis.
• Then, the real integral is extended to a closed path in the complex plane.
• The residue theorem is applied to this closed path as a cycle.
• For this purpose, the residues of the isolated singularities must be determined.

Desired Integral over Part of the Cycle

The integral value needed is actually over a part of the cycle. Therefore, the contribution of the complex line integral, which is added to extend the real line integral into a cycle, must be subtracted from the result of the residue theorem. For improper integrals, there are cases where, in the limit process on the real axis as $+\mathrm{\infty }$ and $-\mathrm{\infty }$ the integral over the added integration path vanishes (approaches 0). In such cases, the desired real integral is equal to the integral over the cycle.

Real Integral as a Line Integral

A real integral is interpreted as a line integral on the real axis as follows. We express the line integral on the real axis as a convex combination of the points $a=a+0i\in ℂ$ and $b=b+0i\in ℂ$ with $a,b\in ℝ$, $a<b$

$\begin{array}{ccc}{\gamma }_{(a,b)}:[a,b]& \to & ℂ\\ t& \mapsto & {\gamma }_{(a,b)}(t)=(1-t)\cdot a+t\cdot b\end{array}$

and

$\underset{a}{\overset{b}{\int }}f(x)dx\text{ mit }a,b\in ℝ$

Improper Integrals as Line Integrals

For improper integrals, a limit process is applied to the integral bounds, e.g.:

$\underset{-\mathrm{\infty }}{\overset{+\mathrm{\infty }}{\int }}f(x)dx=\underset{R\to +\mathrm{\infty }}{\lim }\underset{+R}{\overset{-R}{\int }}f(x)dx$

or

$\underset{0}{\overset{b}{\int }}f(x)dx=\underset{a\to 0,a>0}{\lim }\underset{a}{\overset{b}{\int }}f(x)dx$

Extending to a Cycle

Let ${\gamma }_0:[a,b]\to ℝ$ be the real-valued integration path for which the real integral is to be calculated, and let $G$ be a region with $\mathrm{S}\mathrm{p}\mathrm{u}\mathrm{r}({\gamma }_0)\subset G$. Extend ${\gamma }_0$ to a null-homologous cycle $\mathrm{\Gamma }$ in $G$:

$\mathrm{\Gamma }:={\gamma }_0+\sum _{k=1}^n{n}_k\cdot {\gamma }_k\text{ with }n\in \mathbb{N}\text{ and }{n}_k\in \mathbb{Z}\text{ for all}k\in 1,...,n$

The $n_k$ are usually 1 or -1 if the reversed orientation of the integration path is needed for the cycle extension to $\mathrm{\Gamma }$.

Example 1 - Extending to a Cycle

Let $R\in {ℝ}^{+}$ and ${\gamma }_0:={\gamma }_{(a,b)}$, the integration path from $a\in ℝ$ to $b\in ℝ$ on the real axis as a convex combination. The following extension forms a rectangular path in the complex plane:

$\mathrm{\Gamma }:={\gamma }_{(a,b)}+{\gamma }_{(b,b+iR)}+{\gamma }_{(b+iR,a+iR)}++{\gamma }_{(a+iR,a)}$

Example 2 - Extending to a Cycle

Let $R\in {ℝ}^{+}$ and ${\gamma }_0:={\gamma }_{(+R,-R)}$ on the real axis as a convex combination. The following extension adds an integration path tracing a semicircle with radius $R$ in the complex plane:

$\mathrm{\Gamma }:={\gamma }_{(-R,+R)}+{\gamma }_R$

with

$\begin{array}{ccccc}{\gamma }_R:[0,\pi ]& \to & ℂt& \mapsto & {\gamma }_R(t)=R\cdot e^{it}\end{array}$

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• Date:01/06/2025

de:Kurs:Funktionentheorie/Reelle Integrale mit Residuensatz