contour
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
![## Complex Analysis Integral
### Problem Statement:
Evaluate the following contour integral:
\[ \int_{C} \frac{\sin z}{z^2 - 2iz} \, dz, \]
where the contour \( C \) consists of:
- \( |z| = 3 \) (counterclockwise)
- \( |z| = 1 \) (clockwise)
### Explanation:
This integral problem involves evaluating a complex integral along a given contour consisting of two circles:
1. A counterclockwise path around the circle \(|z| = 3\)
2. A clockwise path around the circle \(|z| = 1\)
The integrand function is \(\frac{\sin z}{z^2 - 2iz}\).
### Analysis:
1. **Singularities**: Determine the singularities of the function \(\frac{\sin z}{z^2 - 2iz}\).
2. **Residue Theorem**: Apply the residue theorem to evaluate the integral around each path.
Analyzing the singularities, we start with the denominator \(z^2 - 2iz\). Factoring it, we have:
\[ z^2 - 2iz = z(z - 2i). \]
So, the singularities \(z = 0\) and \(z = 2i\).
### Steps to Evaluate:
1. **Identify Residues**: Compute the residues of the integrand at the singularities within each contour.
2. **Counterclockwise Contour (|z| = 3)**:
- Lies outside the singularities \(0\) and \(2i\).
3. **Clockwise Contour (|z| = 1)**:
- Lies within the singularity \(z = 0\).
Use this gathered information to conclude the result of the integral using the residue theorem.
###### Note:
When dealing with such problems, make sure to consider the orientation (clockwise vs counterclockwise) as it affects the sign of the integral.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fc6350cd1-610b-4fcb-9d00-d8bcd92f6c10%2F5d2563d6-8f2a-4b54-bd6a-855c4ae5abf0%2Fa7m3rp9_processed.jpeg&w=3840&q=75)
Transcribed Image Text:## Complex Analysis Integral
### Problem Statement:
Evaluate the following contour integral:
\[ \int_{C} \frac{\sin z}{z^2 - 2iz} \, dz, \]
where the contour \( C \) consists of:
- \( |z| = 3 \) (counterclockwise)
- \( |z| = 1 \) (clockwise)
### Explanation:
This integral problem involves evaluating a complex integral along a given contour consisting of two circles:
1. A counterclockwise path around the circle \(|z| = 3\)
2. A clockwise path around the circle \(|z| = 1\)
The integrand function is \(\frac{\sin z}{z^2 - 2iz}\).
### Analysis:
1. **Singularities**: Determine the singularities of the function \(\frac{\sin z}{z^2 - 2iz}\).
2. **Residue Theorem**: Apply the residue theorem to evaluate the integral around each path.
Analyzing the singularities, we start with the denominator \(z^2 - 2iz\). Factoring it, we have:
\[ z^2 - 2iz = z(z - 2i). \]
So, the singularities \(z = 0\) and \(z = 2i\).
### Steps to Evaluate:
1. **Identify Residues**: Compute the residues of the integrand at the singularities within each contour.
2. **Counterclockwise Contour (|z| = 3)**:
- Lies outside the singularities \(0\) and \(2i\).
3. **Clockwise Contour (|z| = 1)**:
- Lies within the singularity \(z = 0\).
Use this gathered information to conclude the result of the integral using the residue theorem.
###### Note:
When dealing with such problems, make sure to consider the orientation (clockwise vs counterclockwise) as it affects the sign of the integral.
Expert Solution
This question has been solved!
Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.
Step by step
Solved in 3 steps with 2 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,
