S. Re{z²}dz; C is the square with corners 0, 1,1+i, i. [1+i]
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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Question
![\[
\int_{C} \text{Re}\{z^2\} \, dz; \text{ C is the square with corners } 0, 1, 1+i, i.
\]
### Explanation:
This expression represents a line integral over a contour \( C \) in the complex plane. The function being integrated is the real part of \( z^2 \), where \( z \) is a complex number. The contour \( C \) is defined as a square with vertices at the complex numbers \( 0 \), \( 1 \), \( 1+i \), and \( i \).
- **Vertices:**
- \( 0 \) corresponds to the point \((0, 0)\).
- \( 1 \) corresponds to the point \((1, 0)\).
- \( 1+i \) corresponds to the point \((1, 1)\).
- \( i \) corresponds to the point \((0, 1)\).
The path starts from \( 0 \) and proceeds in the sequence \( 0 \to 1 \to 1+i \to i \to 0 \), thereby forming a closed square contour in the complex plane. This integral calculates the total accumulation of \(\text{Re}\{z^2\}\) along this path.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F64d8456b-6e21-449f-8cb3-a408fbe9b9b6%2Fa889bbb6-2c77-4f83-b5b0-7787ac9557fd%2F4msdol9_processed.png&w=3840&q=75)
Transcribed Image Text:\[
\int_{C} \text{Re}\{z^2\} \, dz; \text{ C is the square with corners } 0, 1, 1+i, i.
\]
### Explanation:
This expression represents a line integral over a contour \( C \) in the complex plane. The function being integrated is the real part of \( z^2 \), where \( z \) is a complex number. The contour \( C \) is defined as a square with vertices at the complex numbers \( 0 \), \( 1 \), \( 1+i \), and \( i \).
- **Vertices:**
- \( 0 \) corresponds to the point \((0, 0)\).
- \( 1 \) corresponds to the point \((1, 0)\).
- \( 1+i \) corresponds to the point \((1, 1)\).
- \( i \) corresponds to the point \((0, 1)\).
The path starts from \( 0 \) and proceeds in the sequence \( 0 \to 1 \to 1+i \to i \to 0 \), thereby forming a closed square contour in the complex plane. This integral calculates the total accumulation of \(\text{Re}\{z^2\}\) along this path.
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