find the poles for the indicated function. Identify the order of each pole and
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
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![### Identifying Poles and Residues of Given Functions
In this exercise, you are asked to find the poles for the indicated functions \( f(z) \). Additionally, you should identify the order of each pole and the residue of the function at the pole.
#### Problem 4:
\[ f(z) = \frac{\sin z}{z(2z - \pi)} \]
#### Problem 5:
\[ f(z) = \frac{z^2 - 3z + 2}{(z-1)^2(z-3)^3} \]
#### Problem 6:
\[ f(z) = \frac{z + 1}{z^2 + 9} \]
To complete these exercises:
1. Identify the values of \( z \) at which the denominator of the given function becomes zero. These values are the poles of the function.
2. Determine the order of each pole by examining the multiplicity of the factor that causes the numerator to be zero.
3. Calculate the residue at each pole using the appropriate methods, which could include:
- Calculating limits for simple poles.
- For higher-order poles, more advanced techniques such as differentiating the numerator and the denominator may be required.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F4f812bf4-4e37-4075-b786-04672d25a0a0%2Ff69debb9-dbb2-406e-90cb-3f805ae1cebf%2F4iwfbro.jpeg&w=3840&q=75)
Transcribed Image Text:### Identifying Poles and Residues of Given Functions
In this exercise, you are asked to find the poles for the indicated functions \( f(z) \). Additionally, you should identify the order of each pole and the residue of the function at the pole.
#### Problem 4:
\[ f(z) = \frac{\sin z}{z(2z - \pi)} \]
#### Problem 5:
\[ f(z) = \frac{z^2 - 3z + 2}{(z-1)^2(z-3)^3} \]
#### Problem 6:
\[ f(z) = \frac{z + 1}{z^2 + 9} \]
To complete these exercises:
1. Identify the values of \( z \) at which the denominator of the given function becomes zero. These values are the poles of the function.
2. Determine the order of each pole by examining the multiplicity of the factor that causes the numerator to be zero.
3. Calculate the residue at each pole using the appropriate methods, which could include:
- Calculating limits for simple poles.
- For higher-order poles, more advanced techniques such as differentiating the numerator and the denominator may be required.
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