Using the Cauchy Integral Formula for Derivatives, evaluate the following integral 2 +1 dz ++ 2i23
Using the Cauchy Integral Formula for Derivatives, evaluate the following integral 2 +1 dz ++ 2i23
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 4 please
![### Problem 4: Using Cauchy's Integral Formula for Derivatives
Evaluate the following integral:
\[
\oint_{C} \frac{z + 1}{z^4 + 2iz^3} \, dz
\]
#### Hint:
You may find it helpful to rewrite the integrand as:
\[
\frac{z + 1}{z \left( z^3 + 2iz^2 \right)}
\]
The integration is to be performed over the contour \(C\).
### Explanation:
In this problem, you are asked to evaluate a complex contour integral using Cauchy's Integral Formula for derivatives. The integral provided is:
\[
\oint_{C} \frac{z + 1}{z^4 + 2iz^3} \, dz
\]
To simplify this integral, it is suggested that you rewrite the integrand for better insight:
\[
\frac{z + 1}{z \left( z^3 + 2iz^2 \right)}
\]
This hint will help to decompose the integral into a more manageable form by making the relationship between the terms clearer.
When dealing with integrals of this type in complex analysis, Cauchy's Integral Formula and its derivative versions can be highly useful. By rewriting the integrand, you can often identify poles and apply the formula correctly.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F4f812bf4-4e37-4075-b786-04672d25a0a0%2F15eaa9ad-0867-4667-aa07-60560f73d844%2F1rcu8m.jpeg&w=3840&q=75)
Transcribed Image Text:### Problem 4: Using Cauchy's Integral Formula for Derivatives
Evaluate the following integral:
\[
\oint_{C} \frac{z + 1}{z^4 + 2iz^3} \, dz
\]
#### Hint:
You may find it helpful to rewrite the integrand as:
\[
\frac{z + 1}{z \left( z^3 + 2iz^2 \right)}
\]
The integration is to be performed over the contour \(C\).
### Explanation:
In this problem, you are asked to evaluate a complex contour integral using Cauchy's Integral Formula for derivatives. The integral provided is:
\[
\oint_{C} \frac{z + 1}{z^4 + 2iz^3} \, dz
\]
To simplify this integral, it is suggested that you rewrite the integrand for better insight:
\[
\frac{z + 1}{z \left( z^3 + 2iz^2 \right)}
\]
This hint will help to decompose the integral into a more manageable form by making the relationship between the terms clearer.
When dealing with integrals of this type in complex analysis, Cauchy's Integral Formula and its derivative versions can be highly useful. By rewriting the integrand, you can often identify poles and apply the formula correctly.
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