---
title: "Evaluating Real Integrals by Residues"
description: "Close a real integral into a contour in the plane, discard the added arc with an estimate, and read the answer off the residues."
canonical: https://lightmysky.com/learn/mathematics/evaluating-real-integrals-by-residues-mt_Lj_HXtjzAh
source: https://lightmysky.com/learn/mathematics/evaluating-real-integrals-by-residues-mt_Lj_HXtjzAh.md
retrieved: 2026-09-12
---

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# Evaluating Real Integrals by Residues

Close a real integral into a contour in the plane, discard the added arc with an estimate, and read the answer off the residues.

Subject: Mathematics · Area: Complex Analysis · Ages 21 to 22
Page: https://lightmysky.com/learn/mathematics/evaluating-real-integrals-by-residues-mt_Lj_HXtjzAh

## Ready when they can

- Evaluate a rational real integral over the whole line by closing in a half-plane
- Handle an oscillatory integrand with Jordan's lemma and justify discarding the arc
- Deal with a pole on the contour by indenting around it and accounting for the half residue

## Lesson: Close the line and read residues

Real integrals over the whole line become loop integrals by adding a giant semicircle. For rational functions decaying fast enough, the arc share dies as the radius grows, leaving 2 pi i times the upper half plane residues. Decay is the whole game: the denominator must outgrow the numerator by at least two degrees. Check this before closing.

Sines and cosines need Jordan lemma. Write the cosine as the real part of e to the i a z, close upwards for positive frequency, and the upstairs exponential decay kills the arc. Take the real part at the end. The sign picks the half plane: positive frequencies close up, negative ones close down. The wrong side makes the arc explode.

Poles sitting on the path need indenting. Detour around the pole with a tiny semicircle, which in the limit contributes half a residue, positive or negative by the detour side. The straight runs become a principal value integral. Sketch the detour and its orientation with care, since one sign slip ruins the sum.

**Tip.** Close with intent, not hope. Name the half plane, verify the decay, list enclosed poles, then treat boundary poles by indenting. Never count a pole you did not enclose.

**Recap.** Close the line into a loop, kill the arc, and let enclosed residues answer.

## Practice

17 questions on this page, each with its working shown.

## Needs first

- [The Residue Theorem](https://lightmysky.com/learn/mathematics/the-residue-theorem-mt_BN2biZ_zos)
- [Improper Integrals and Their Convergence](https://lightmysky.com/learn/mathematics/improper-integrals-and-their-convergence-mt_gvtZPFRA6Z)

## Opens up

- [The Argument Principle and Rouche's Theorem](https://lightmysky.com/learn/mathematics/the-argument-principle-and-rouches-theorem-mt_J1dC428M84)