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 $+\infty$ and $-\infty$ 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 $a=a+0i\in \mathbb {C}$ and $b=b+0i\in \mathbb {C}$ with $a,b\in \mathbb {R}$, $a<b$

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

and

$\displaystyle \int _{a}^{b}f(x)dx{\mbox{ mit }}a,b\in \mathbb {R}$

Improper Integrals as Line Integrals

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

$\displaystyle \int _{-\infty }^{+\infty }f(x)dx=\displaystyle \lim _{R\to +\infty }\int _{+R}^{-R}f(x)dx$

or

$\displaystyle \int _{0}^{b}f(x)dx=\displaystyle \lim _{a\to 0,a>0}\int _{a}^{b}f(x)dx$

Extending to a Cycle

Let $\gamma _{0}:[a,b]\to \mathbb {R}$ be the real-valued integration path for which the real integral is to be calculated, and let $G$ be a region with $\mathrm {Spur} (\gamma _{0})\subset G$. Extend $\gamma _{0}$ to a null-homologous cycle $\Gamma$ in $G$:

$\Gamma :=\gamma _{0}+\sum _{k=1}^{n}n_{k}\cdot \gamma _{k}{\mbox{ with }}n\in \mathbb {N} {\mbox{ and }}n_{k}\in \mathbb {Z} {\mbox{ for all}}k\in {1,...,n}$

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

Example 1 - Extending to a Cycle

Let $R\in \mathbb {R} ^{+}$ and $\gamma _{0}:=\gamma _{(a,b)}$, the integration path from $a\in \mathbb {R}$ to $b\in \mathbb {R}$ on the real axis as a convex combination. The following extension forms a rectangular path in the complex plane:

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

Example 2 - Extending to a Cycle

Let $R\in \mathbb {R} ^{+}$ 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:

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

with

${\begin{array}{rcl}\gamma _{R}:[0,\pi ]&\to &\mathbb {C} \ t&\mapsto &\gamma _{R}(t)=R\cdot e^{it}\ \end{array}}$

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• Date:01/06/2025