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Integrating the Poisson Kernel

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I have been trying several methods to try to integrate the following function:
*Recall that Pa(theta) = (1-a^(2))/(1-2a*cos(theta) + a^(2)) and see the attached file for the proper notation.

I substituted 1/2(z+1/z) for cos (theta) and reduced the numerator to (az-1)(a-z) in hopes that there would be some type of cancelation to get it to an integral I can solve. I am looking for guidance on how to proceed.

Thank you.

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Solution Summary

This solution integrates the given function.

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We are going to transform the given integral into a line integral on the circle | z | = 1, using certain ...

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