cosh 2 dz
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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2b please. The boundary should be x=+ and - 2 and y= + or - 2
![### Problem Statement
2. Let \( C \) be the positively oriented boundary of the square whose sides lie along the lines \( x = \pm 1 \) and \( x = \pm 2 \). Evaluate the following integrals:
(a)
\[
\int_{C} \frac{z}{z^2 + 1} \, dz
\]
(b)
\[
\int_{C} \frac{\cosh z}{z^2 + 3} \, dz
\]
### Explanation
Here we are given a contour \( C \), which is the positively oriented boundary of a square. The sides of the square lie along the lines \( x = \pm 1 \) and \( x = \pm 2 \). Along this contour, we are required to evaluate two integrals:
**Integral (a):**
The integral
\[
\int_{C} \frac{z}{z^2 + 1} \, dz
\]
involves the integrand \(\frac{z}{z^2 + 1}\), which is to be integrated along the contour \( C \).
**Integral (b):**
The integral
\[
\int_{C} \frac{\cosh z}{z^2 + 3} \, dz
\]
involves the integrand \(\frac{\cosh z}{z^2 + 3}\), which is to be integrated along the same contour \( C \).
### Mathematical Background
In complex analysis, evaluating contour integrals often involves using techniques such as the residue theorem. We look for singularities inside the contour \( C \) and calculate residues at those points to find the value of the integral. The given integrals suggest that we might need to identify poles and apply the residue theorem to solve them.
For educational purposes, students should:
1. Identify possible singularities of the integrands \(\frac{z}{z^2 + 1}\) and \(\frac{\cosh z}{z^2 + 3}\).
2. Determine which of these singularities lie inside the given contour \( C \).
3. Calculate residues at these points.
4. Apply the residue theorem to evaluate the integrals.
This problem serves as an exercise in understanding and applying concepts of contour integration and residue calculus in complex analysis.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F4f812bf4-4e37-4075-b786-04672d25a0a0%2Fd8a5b8e3-5282-4b17-84f6-1b9f76d06ed5%2F7c8wcso.jpeg&w=3840&q=75)
Transcribed Image Text:### Problem Statement
2. Let \( C \) be the positively oriented boundary of the square whose sides lie along the lines \( x = \pm 1 \) and \( x = \pm 2 \). Evaluate the following integrals:
(a)
\[
\int_{C} \frac{z}{z^2 + 1} \, dz
\]
(b)
\[
\int_{C} \frac{\cosh z}{z^2 + 3} \, dz
\]
### Explanation
Here we are given a contour \( C \), which is the positively oriented boundary of a square. The sides of the square lie along the lines \( x = \pm 1 \) and \( x = \pm 2 \). Along this contour, we are required to evaluate two integrals:
**Integral (a):**
The integral
\[
\int_{C} \frac{z}{z^2 + 1} \, dz
\]
involves the integrand \(\frac{z}{z^2 + 1}\), which is to be integrated along the contour \( C \).
**Integral (b):**
The integral
\[
\int_{C} \frac{\cosh z}{z^2 + 3} \, dz
\]
involves the integrand \(\frac{\cosh z}{z^2 + 3}\), which is to be integrated along the same contour \( C \).
### Mathematical Background
In complex analysis, evaluating contour integrals often involves using techniques such as the residue theorem. We look for singularities inside the contour \( C \) and calculate residues at those points to find the value of the integral. The given integrals suggest that we might need to identify poles and apply the residue theorem to solve them.
For educational purposes, students should:
1. Identify possible singularities of the integrands \(\frac{z}{z^2 + 1}\) and \(\frac{\cosh z}{z^2 + 3}\).
2. Determine which of these singularities lie inside the given contour \( C \).
3. Calculate residues at these points.
4. Apply the residue theorem to evaluate the integrals.
This problem serves as an exercise in understanding and applying concepts of contour integration and residue calculus in complex analysis.
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