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Complex Integration : Holomorphisms

Let f(z) be holomorphic on the unit disc and f(0)=1. By working with
1/2ipi(integral over unit circle of [2+,-(z+1/z)]f(z) dz/z) prove that
a)2/pi(integral(0 -2pi) of f(e^itheta)cos^2theta/2 d(theta))=2 + f'(0)
b)2/pi(integral(0-2pi) of f(e^itheta)sin^2theta/2 d(theta))=2-f'(0)

Solution Summary

Holomorphisms are investigated through integration. The solution is detailed and well presented. The response was given a rating of "5/5" by the student who originally posted the question.