Let C, be the straight line path z = (1 + í)4, 0 sts 1; Ca the quarter circle z = 1 - cos t + i sin t, 0
Let C, be the straight line path z = (1 + í)4, 0 sts 1; Ca the quarter circle z = 1 - cos t + i sin t, 0
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
Related questions
Question
Just number 8
![The text outlines a problem involving the evaluation of complex line integrals along different paths. The situation described uses three paths in the complex plane:
- **C₁**: A straight line path, \( z = (1 + t)i \), where \( 0 \leq t \leq 1 \).
- **C₂**: A quarter circle path, \( z = 1 - \cos t + i \sin t \), where \( 0 \leq t \leq \frac{\pi}{2} \).
- **C₃**: A path defined by \( z = t \), where \( 0 \leq t \leq 1 \), and \( z = 1 + i(t-1) \), where \( 1 \leq t \leq 2 \).
The task is to demonstrate the following expressions:
7. \[
\int_{C_1} z \, dz = \int_{C_2} z \, dz = \int_{C_3} z \, dz = i
\]
8. \[
\int_{C_1} z^* \, dz = 1, \quad \int_{C_2} z \, dz = 1 + i(1 - \frac{\pi}{2}), \quad \int_{C_3} z \, dz = 1 + i
\]
In this problem, \( z^* \) denotes the complex conjugate of \( z \). Each integral represents a line integral of the complex function along the given path.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F64d8456b-6e21-449f-8cb3-a408fbe9b9b6%2F91896230-03dc-4867-9ed7-5107973cdc58%2Fipiqald_processed.png&w=3840&q=75)
Transcribed Image Text:The text outlines a problem involving the evaluation of complex line integrals along different paths. The situation described uses three paths in the complex plane:
- **C₁**: A straight line path, \( z = (1 + t)i \), where \( 0 \leq t \leq 1 \).
- **C₂**: A quarter circle path, \( z = 1 - \cos t + i \sin t \), where \( 0 \leq t \leq \frac{\pi}{2} \).
- **C₃**: A path defined by \( z = t \), where \( 0 \leq t \leq 1 \), and \( z = 1 + i(t-1) \), where \( 1 \leq t \leq 2 \).
The task is to demonstrate the following expressions:
7. \[
\int_{C_1} z \, dz = \int_{C_2} z \, dz = \int_{C_3} z \, dz = i
\]
8. \[
\int_{C_1} z^* \, dz = 1, \quad \int_{C_2} z \, dz = 1 + i(1 - \frac{\pi}{2}), \quad \int_{C_3} z \, dz = 1 + i
\]
In this problem, \( z^* \) denotes the complex conjugate of \( z \). Each integral represents a line integral of the complex function along the given path.
Expert Solution
This question has been solved!
Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.
This is a popular solution!
Trending now
This is a popular solution!
Step by step
Solved in 4 steps with 3 images
Recommended textbooks for you
Advanced Engineering Mathematics
Advanced Math
ISBN:
9780470458365
Author:
Erwin Kreyszig
Publisher:
Wiley, John & Sons, Incorporated
Numerical Methods for Engineers
Advanced Math
ISBN:
9780073397924
Author:
Steven C. Chapra Dr., Raymond P. Canale
Publisher:
McGraw-Hill Education
Introductory Mathematics for Engineering Applicat…
Advanced Math
ISBN:
9781118141809
Author:
Nathan Klingbeil
Publisher:
WILEY
Advanced Engineering Mathematics
Advanced Math
ISBN:
9780470458365
Author:
Erwin Kreyszig
Publisher:
Wiley, John & Sons, Incorporated
Numerical Methods for Engineers
Advanced Math
ISBN:
9780073397924
Author:
Steven C. Chapra Dr., Raymond P. Canale
Publisher:
McGraw-Hill Education
Introductory Mathematics for Engineering Applicat…
Advanced Math
ISBN:
9781118141809
Author:
Nathan Klingbeil
Publisher:
WILEY
Mathematics For Machine Technology
Advanced Math
ISBN:
9781337798310
Author:
Peterson, John.
Publisher:
Cengage Learning,
