Polar Coordinates to evaluate integrals

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Homework Help Overview

The discussion revolves around evaluating integrals using polar coordinates, specifically in the context of a unit circle in the complex plane. The original poster expresses uncertainty about the application of polar integrals and the residue theorem.

Discussion Character

  • Exploratory, Conceptual clarification, Mathematical reasoning

Approaches and Questions Raised

  • The original poster attempts to relate polar integrals to their previous experiences with sine and cosine functions and residue calculations. Some participants suggest specific substitutions and reference the Cauchy integral formula, while others question the correctness of the final expression derived by the original poster.

Discussion Status

The discussion is ongoing, with participants exploring different interpretations of the problem and referencing relevant mathematical concepts. Some guidance has been provided regarding the Cauchy integral formula, but there is no explicit consensus on the approach to take.

Contextual Notes

There is an indication that the original poster may be seeking to consolidate their understanding of theory through practice questions, suggesting a focus on learning and application rather than immediate resolution.

Fixxxer125
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Homework Statement


Use Polar coordinates to evaluate
2eydkky.png
were C denotes the unit circle about a fixed point Z0 in the complex plane



The Attempt at a Solution


I've only used polar integrals to convert an integral in sin and cos into one in therms of z, find the residues and then use the residue theorum to evaluate the integral so I am not really sure where to go with this question? Any help would be greatly appreciated!
 
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Let [itex]z= z_0+ e^{i\theta}[/itex].
 
Have I done it correctly if I end up with a final answer of
2∏i(aZ02 + bZ0 + c)
Thanks!
 
Yes, in fact there is the "Cauchy integral formula" that says
[tex]\oint \frac{f(z)}{z-z_0} dz= 2\pi if(z_0)[/tex]

Perhaps this problem was intended as an introduction to that.
 
Ah yes we have done that previously, I think I just need to do practise questions to bring all the theory together. Cheers
 

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