# Evaluating an Integral with 2nd Order Pole : Moivre-Laplace Fomulation

Problem: Evaluate the integral from 0 to INF of: (x^a)/(x^2 +4)^2 dx, -1 < a < 3 We are to use f(z)= (z^a)/(z^2 +4)^2, with z^a = e^(a Log z), Log z= ln|z| + i Arg z, and -pi/2 < Arg z < 3pi/2. I have found the residue at 2i to be: [2^a(1-a)/16]*[cos ((pi*a)/2) + i sin ((pi*a)/2). Please let me know if this is correct and how to solve this problem. Many thanks for your help.

Use the branch cut of the negative Im axis, and use the following curve:

C= -R to R, -p to p, p to R, and Cr from 0 to pi.

Can anyone do this problem using these parameters?

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#### Solution Preview

The integral in question is

; where -1 < a < 3.

Corresponding function of complex variable is f(z) = . It should be integrated over the closed contour: [- R; -r] ïƒˆ Cr ïƒˆ[ r; R] ïƒˆ CR ; where Cr - semi-circle of small radius r, and CR - semi-circle of large radius R in the upper part of the complex plane, with the ...

#### Solution Summary

An integral is found using a Moivre-Laplace formulation. The solution is detailed and well presented.