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Complex Analysis - Contour integration

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Using contour integral methods in the complex plane and the residue theorem.

see attached

Y please give me the full "Y" treatment on this as I want to really understand it Thanks C

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The expert examines the complex analysis for contour integration.

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The integral is:
The integration is over the real axis.
Note that
We can write it as a complex integral:

In order to take advantage of Cauchy's residue theorem, we would like to find a closed path that includes the real axis and some path that we know the value of the integral on.
Let's look at the first integral
If the real axis is the path and the other yet unknown path is then the closed path is and:

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