Find the Cauchy principal value Lant dx (x+1)(x²+2) -R of the following integral R
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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![Find the Cauchy principal value of the following integral:
\[
\int_{-\infty}^{\infty} \frac{dx}{(x+1)(x^2+2)}
\]
### Diagram Explanation:
The diagram illustrates a contour in the complex plane, which is used to evaluate the given integral using the method of residues. The path consists of four segments:
- **\(C_1\):** A semicircular arc in the upper half of the complex plane with radius \(R\), centered at the origin. The arc is oriented counterclockwise.
- **\(C_2\):** A horizontal line segment from \(-R\) to the origin along the x-axis.
- **\(C_3\):** A small semicircular arc around the pole at the origin, with radius approaching zero. This arc is oriented clockwise.
- **\(C_4\):** A line segment from the origin to \(R\) along the x-axis.
The notation \(i\sqrt{2}\) indicates a point on the imaginary axis, corresponding to one of the singularities of the integrand.
This contour setup is used to evaluate the integral by considering the residues at the poles within the contour, thus allowing the application of Cauchy's Residue Theorem.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F71f6a4b8-527f-4030-b85a-828002701c6f%2F4bfae8aa-3645-4fe3-ad7a-8fd13c7d6d7d%2Foyxt8rr_processed.png&w=3840&q=75)
Transcribed Image Text:Find the Cauchy principal value of the following integral:
\[
\int_{-\infty}^{\infty} \frac{dx}{(x+1)(x^2+2)}
\]
### Diagram Explanation:
The diagram illustrates a contour in the complex plane, which is used to evaluate the given integral using the method of residues. The path consists of four segments:
- **\(C_1\):** A semicircular arc in the upper half of the complex plane with radius \(R\), centered at the origin. The arc is oriented counterclockwise.
- **\(C_2\):** A horizontal line segment from \(-R\) to the origin along the x-axis.
- **\(C_3\):** A small semicircular arc around the pole at the origin, with radius approaching zero. This arc is oriented clockwise.
- **\(C_4\):** A line segment from the origin to \(R\) along the x-axis.
The notation \(i\sqrt{2}\) indicates a point on the imaginary axis, corresponding to one of the singularities of the integrand.
This contour setup is used to evaluate the integral by considering the residues at the poles within the contour, thus allowing the application of Cauchy's Residue Theorem.
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