dz -i|=1 (22+i) "[( -- 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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![The image contains a mathematical expression involving a contour integral, expressed as:
\[ \oint_{|z-i|=1} \frac{dz}{z^2+i} \left[ \frac{\pi}{\sqrt{2}} (1-i) \right]. \]
The integral is taken over the contour where \(|z-i| = 1\), which describes a circle in the complex plane centered at \(z = i\) with a radius of 1.
### Explanation of Components:
- **Contour Integral**: The symbol \(\oint\) indicates a contour integral, a common concept in complex analysis used to evaluate integrals along paths in the complex plane.
- **Complex Function**: The integrand is \(\frac{dz}{z^2 + i}\), which is a rational function in the complex variable \(z\).
- **Multiplier**: The contents inside the brackets, \(\left[ \frac{\pi}{\sqrt{2}} (1-i) \right]\), suggests multiplication by a complex constant. The constant involves both real and imaginary components, and includes \(\pi\) divided by the square root of 2.
This expression is often encountered in the context of evaluating complex integrals using techniques such as the residue theorem, particularly when the poles of the function lie inside the contour of integration.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F64d8456b-6e21-449f-8cb3-a408fbe9b9b6%2F5f33b382-d128-4bcb-b5fc-bcd2e4a2e7e2%2Fky9wdqa_processed.png&w=3840&q=75)
Transcribed Image Text:The image contains a mathematical expression involving a contour integral, expressed as:
\[ \oint_{|z-i|=1} \frac{dz}{z^2+i} \left[ \frac{\pi}{\sqrt{2}} (1-i) \right]. \]
The integral is taken over the contour where \(|z-i| = 1\), which describes a circle in the complex plane centered at \(z = i\) with a radius of 1.
### Explanation of Components:
- **Contour Integral**: The symbol \(\oint\) indicates a contour integral, a common concept in complex analysis used to evaluate integrals along paths in the complex plane.
- **Complex Function**: The integrand is \(\frac{dz}{z^2 + i}\), which is a rational function in the complex variable \(z\).
- **Multiplier**: The contents inside the brackets, \(\left[ \frac{\pi}{\sqrt{2}} (1-i) \right]\), suggests multiplication by a complex constant. The constant involves both real and imaginary components, and includes \(\pi\) divided by the square root of 2.
This expression is often encountered in the context of evaluating complex integrals using techniques such as the residue theorem, particularly when the poles of the function lie inside the contour of integration.
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