Let C, be the straight line path z = (1 + i)t, 0 Sts 1; Cz the quarter circle z =1- cost +isin t, 0 ses }; and Ca the path ==t, (0SIS 1), 1+i(t – 1) (1
Let C, be the straight line path z = (1 + i)t, 0 Sts 1; Cz the quarter circle z =1- cost +isin t, 0 ses }; and Ca the path ==t, (0SIS 1), 1+i(t – 1) (1
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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![**Integral Paths and Calculation**
Let \( C_1 \) be the straight line path \( z = (1+i)t \), \( 0 \leq t \leq 1 \); \( C_2 \) the quarter circle \( z = 1 - \cos t + i \sin t \), \( 0 \leq t \leq \frac{\pi}{2} \); and \( C_3 \) the path \( z = t \), \( (0 \leq t \leq 1) \), \( 1 + i(t-1) \) \( (1 \leq t \leq 2) \).
Show that:
\[
\int_{C_1} z \, dz = \int_{C_2} z \, dz = \int_{C_3} z \, dz = i.
\]
**Explanation:**
The problem involves evaluating complex line integrals over specified paths in the complex plane.
- **\( C_1 \)**: A straight line path defined by the equation \( z = (1+i)t \) with parameter \( t \) ranging from 0 to 1.
- **\( C_2 \)**: A quarter circle traced out by the complex exponential form, ranging from \( t = 0 \) to \( t = \frac{\pi}{2} \).
- **\( C_3 \)**: This path combines two segments. The first part, \( z = t \), runs from \( t = 0 \) to \( t = 1 \), and the second segment follows \( 1 + i(t-1) \) from \( t = 1 \) to \( t = 2 \).
The task is to evaluate and confirm that the integral of \( z \) over each of these paths results in \( i \).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F64d8456b-6e21-449f-8cb3-a408fbe9b9b6%2F2409a899-039b-4269-8aad-ef95728ffc7f%2F8cwomjm_processed.png&w=3840&q=75)
Transcribed Image Text:**Integral Paths and Calculation**
Let \( C_1 \) be the straight line path \( z = (1+i)t \), \( 0 \leq t \leq 1 \); \( C_2 \) the quarter circle \( z = 1 - \cos t + i \sin t \), \( 0 \leq t \leq \frac{\pi}{2} \); and \( C_3 \) the path \( z = t \), \( (0 \leq t \leq 1) \), \( 1 + i(t-1) \) \( (1 \leq t \leq 2) \).
Show that:
\[
\int_{C_1} z \, dz = \int_{C_2} z \, dz = \int_{C_3} z \, dz = i.
\]
**Explanation:**
The problem involves evaluating complex line integrals over specified paths in the complex plane.
- **\( C_1 \)**: A straight line path defined by the equation \( z = (1+i)t \) with parameter \( t \) ranging from 0 to 1.
- **\( C_2 \)**: A quarter circle traced out by the complex exponential form, ranging from \( t = 0 \) to \( t = \frac{\pi}{2} \).
- **\( C_3 \)**: This path combines two segments. The first part, \( z = t \), runs from \( t = 0 \) to \( t = 1 \), and the second segment follows \( 1 + i(t-1) \) from \( t = 1 \) to \( t = 2 \).
The task is to evaluate and confirm that the integral of \( z \) over each of these paths results in \( i \).
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