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shehry1
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Homework Statement
[tex] lim_{\alpha\rightarrow 0} \int_{-\infty}^{\infty} dx. e^{ixy}/(2\Pi i (x-i \alpha)) = H(y)[/tex]
where H(y) is the step function ie. H(y) = 1 for y > 0, H(y) = 0 (otherwise)
Compute using an appropriate contour integral.
Homework Equations
-Laurent series
-Residue theorem
-Concepts from contour integration
The Attempt at a Solution
I found the residue and its [tex] e^{-y \alpha} [/tex] for a semi circular contour lying in the top half plane. Mathematica vouches for that.
To find the integral, I followed the usual procedure.
1. Define a semi circular contour
2. Separate the contour into two parts one along the x-axis and the other constituting the semicircular arc. The straight line becomes the desired integral.
Problems:
-How do I reduce the Integral for the semi-circular arc to zero.
-If it IS zero, how do I relate the residue which is an exponential to the step function.
I was thinking that this might be something like an inverse Fourier transform. But the Fourier transform of the Step function involves a Delta function: http://mathworld.wolfram.com/FourierTransformHeavisideStepFunction.html