I need another easier answer for this question

timer Asked: Nov 16th, 2015
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Question Description

I need another easier answer for this question

Tutor Answer

School: UC Berkeley

Thank you for the opportunity to help you with your question!

We will use the semicircle as the contour of integration. Just for simplicity, to justify this integration, the integral on the arc going from : π to -π, vanishes, so I only need to evaluate the integral on the x-axis, i.e. real line:

The function of choice should be : 

I need to find the roots of the following equation:Let z = (re^iθ)

write -1 as: -1 = e^(iπ +i2πn), (z^4) = = -1, becomes: (r^4)(e^(i4θ)) = (e^i(π+4nπ))

Equate magnitude to magnitude, to get : r^4 = 1, hence r = 1.

Equate argument to argument to get: , let n = 0,1,2,3 to get all the roots:

They are: 

Let the zeros be denoted: 

Rewrite f(z) as followed: f(z) = 

Only the first two zero are inside the semicircle:

The residue at , the math is messy, but it's:

The residue at 

Then 2πi times by the sum of the residue at z1 and z2 gives you the desired answer:

This is basically Cauchy's integral formula, which is a special case of the Residue theorem.

Please let me know if you need any clarification. I'm always happy to answer your questions.

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