- #1
stephenkeiths
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Homework Statement
I'm trying to evaluate the integral
[itex]I(a)=\int\frac{cos(ax)}{x^{4}+1}[/itex]
from 0 to ∞
Homework Equations
To do this I'm going to consider the complex integral:
[itex]J=\oint\frac{e^{iaz}}{z^{4}+1}[/itex]
Over a semi-circle of radius R in the upper half plane, then let R-->∞
There are 2 curves, the straight curve along the real axis, and the semi-circular arc.
The Attempt at a Solution
Along the first curve (along the real axis) the line integral is just
[itex]I_{1}=\int\frac{e^{iax}}{x^{4}+1}[/itex]
from 0 to infinity
Now the real part of this is 2*I(a)
The second line integral goes to zero. So 2*I(a)=Re(J).
So I need to evaluate J by residue calculus and take the real part to get I(a).
So there are 2 simple poles inside of the semi-circle, namely
[itex]z=e^{i\frac{\pi}{4}}[/itex] and [itex]z=e^{i\frac{3\pi}{4}}[/itex]
So [itex]J=2i\pi(Res(f,e^{i\frac{\pi}{4}})+Res(f,e^{i\frac{3\pi}{4}}))[/itex]
But I'm having trouble computing the Residues.
I know that [itex]Res(f,e^{i\frac{\pi}{4}})=\frac{e^{iaz}}{4z^{3}} z=e^{i\frac{\pi}{4}}[/itex]
But I'm struggling with the exponential to an exponential, I can't get a "clean" answer. I'm not sure if there's a better way I should go about this. Or if there's something I'm doing wrong.
Please Help!