de Verify by using Residue theorem. /3 2+cos 0
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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need help with residue theorem for complex analysis
![**Problem Statement:**
Verify the integral
\[
\int_{0}^{\pi} \frac{d\theta}{2 + \cos\theta} = \frac{\pi}{\sqrt{3}}
\]
by using the Residue Theorem.
**Explanation:**
This problem involves evaluating a definite integral using the Residue Theorem, a powerful tool in complex analysis. The integral is expressed as a function of \( \theta \), where the cosine function \( \cos\theta \) is present in the denominator. The integral is evaluated over the interval from 0 to \( \pi \).
The goal is to confirm that the integral equals \( \frac{\pi}{\sqrt{3}} \) by applying techniques from complex analysis, particularly leveraging the Residue Theorem to handle integrals that can be expressed in terms of complex functions.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F0548f28d-2867-4ce8-91e2-809a65372be3%2F2a5097ca-c5f0-4013-b8cd-4a3df6d4e949%2F70ey6jo_processed.png&w=3840&q=75)
Transcribed Image Text:**Problem Statement:**
Verify the integral
\[
\int_{0}^{\pi} \frac{d\theta}{2 + \cos\theta} = \frac{\pi}{\sqrt{3}}
\]
by using the Residue Theorem.
**Explanation:**
This problem involves evaluating a definite integral using the Residue Theorem, a powerful tool in complex analysis. The integral is expressed as a function of \( \theta \), where the cosine function \( \cos\theta \) is present in the denominator. The integral is evaluated over the interval from 0 to \( \pi \).
The goal is to confirm that the integral equals \( \frac{\pi}{\sqrt{3}} \) by applying techniques from complex analysis, particularly leveraging the Residue Theorem to handle integrals that can be expressed in terms of complex functions.
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