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Residual calculus

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Use residue theory to compute the integral:

Use residue theory to compute the integral:

I know that the singular points are 4i and -4i, and that I need to integrate from -R to R and then around the half circle from 0 to pi. Only 4i is in this half-circle, so I need to calculate the residue for 4i. This is where I get stuck and then I need to know how to put it all together at the end. Please advise, and show each step. Thanks!

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This shows how to use residue theory to compute the integral. The expert calculates the residues for half-circles.

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Let's consider the integral , where and is consist of the half circle and the line segment from ...

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