- #1
ijmbarr
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Homework Statement
I am trying to evaluate the integral:
[tex]
I = \int^{\infty}_{-\infty} \frac{z e^{irz}\ }{(z-k)(z+k)} dz
[/tex]
The way I attepted it was to use contour integration around a semicircle in the top half of the argan diagram, with two small indents above the poles. This means that the contour doesn't contain either pole, but has contributions from the indents around them.
I calculated the residue at each pole to be:
[tex]
R(k) = e^{irk}/2
[/tex]
[tex]
R(-k) = e^{-irk}/2
[/tex]
Using the method here: http://en.wikipedia.org/wiki/Residue_(complex_analysis)#Calculating_residues
From jordans lemma, the contribution from the semicircle tends to zero as its radius tends to infinity, and the small indents around each pole contribute [tex]- i\! \pi \! residue[/tex]. The overall intergral should be:
[tex]
I = i \pi (e^{irk}\ + e^{-irk})/2
[/tex]
However the answers I have say that the intergral is
[tex]
I = i \pi e^{irk}
[/tex]
Is there an error somewhere that I'm missing? Does it depend where you choose the contour to be?
Homework Equations
The Attempt at a Solution
[Edited because of typos]
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