# Residue Theorem Application

1. Aug 19, 2008

### Feanor

Hi everyone.
I'm a brazilian mathmatician that didn't studied complex analysis. I study finance and now I'm needing to study that.
In a paper of Lewis (2001) I found an expression that I couldn't understand.
Does anyone can help me with that? They say they use the Residue theorem but I couldn't make the calculations using the versions of this theorem that I found.
The equality is the following:

$\int_{i Im(u)-\infty} ^{i Im(u)+ \infty} \left( \int_{0} ^{\infty} e^{iuA_t} \Phi^{\ast}(u)dx \right) du= \pi + 2 \left( \int_{0} ^{\infty} Re \left[ \frac{e^{-iulnK} \Phi^{\ast}(-u) } {iu}\right] du \right)$
(jpg attached for non tex users)
Could you send me reference that I could read and understand the above?

Thanks!

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2. Aug 21, 2008

### Feanor

Come on people...
Any suggestions on how to solve it?
I basically need a relation between a contour integral and the real part of another one.

Thanks!