Advanced Engineering Mathematics
10th Edition
ISBN: 9780470458365
Author: Erwin Kreyszig
Publisher: Wiley, John & Sons, Incorporated
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![**Title: Evaluating Integrals Using Residues**
**Problem Statement:**
Using residues, evaluate the integral
\[
\int_{-\infty}^{\infty} \frac{4}{(x^2 + 1)^3} \, dx
\]
**Explanation:**
This problem involves the use of complex analysis techniques to evaluate a real integral by means of contour integration and residue theory. The integrand \(\frac{4}{(x^2 + 1)^3}\) suggests singularities in the complex plane, which can be analyzed to find the residue that contributes to the integral value over the real line. The focus is on identifying singularities within the contour and calculating residues to simplify the evaluation process.
**Key Concepts:**
- **Residue Theory:** A method in complex analysis to evaluate integrals by examining the behavior of functions near their singularities.
- **Contour Integration:** A technique used to evaluate complex integrals, often by means of a closed contour in the complex plane.
Readers interested in learning more should explore topics such as complex functions, poles, and the residue theorem, which are integral to understanding these advanced mathematical concepts.](https://content.bartleby.com/qna-images/question/a62f7b35-7db6-46d8-92c3-a45ad2747ea7/f3c47eda-2321-4cd4-80bd-3dbe0167afb0/43mewn_processed.png)
Transcribed Image Text:**Title: Evaluating Integrals Using Residues**
**Problem Statement:**
Using residues, evaluate the integral
\[
\int_{-\infty}^{\infty} \frac{4}{(x^2 + 1)^3} \, dx
\]
**Explanation:**
This problem involves the use of complex analysis techniques to evaluate a real integral by means of contour integration and residue theory. The integrand \(\frac{4}{(x^2 + 1)^3}\) suggests singularities in the complex plane, which can be analyzed to find the residue that contributes to the integral value over the real line. The focus is on identifying singularities within the contour and calculating residues to simplify the evaluation process.
**Key Concepts:**
- **Residue Theory:** A method in complex analysis to evaluate integrals by examining the behavior of functions near their singularities.
- **Contour Integration:** A technique used to evaluate complex integrals, often by means of a closed contour in the complex plane.
Readers interested in learning more should explore topics such as complex functions, poles, and the residue theorem, which are integral to understanding these advanced mathematical concepts.
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