Hi everyone, I have a big test tomorrow and I'm still hung up on integrating over branch cuts.
So I was wondering how to show
where
Consider the integral,
If we let then we get,
Therefore, to evaluate your integral is is sufficient to evaluate the first (more convient) integral.
Fix (and "large" i.e. large enough for all the inequalities to work out)
let be the rectangular contour with vertices .
Now define the function the function has a pole at with .
Integrating by the residue theorem we get that,
We will now show that the 2nd and 4th integrals (over the vertical sides) go to zero as .
Look at the function inside the 2nd integral,
Look at the function inside the 4th integral,
Therefore the 2nd integral satisfies,
Therefore the 4th integral satisfies,
Look at the 3rd integral,
Taking the limits in the big equation with four integral above we get,
Finally,
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At this point we can derive the Euler Reflection Formula.
Remember by the first comment we have shown that,
Let to get,
Hello,
Lemma : (actually, I only know the French naming of this...)
where R is a rational fraction without poles on the real axis and .
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
This can be proved by considering the contour integral :
over this contour :
as
Let
When t goes to infinity, . Then
. When z is near from 0, we thus have
The comes from the fact that :
But
Hence, by the residue theorem, we have :
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
There is only one pole over :
We have
Hence
By the above lemma, we get :
Hence :
Okay, there may be mistakes and I'm sure there are unclear things, but one can just remember the formula of the lemma. Sorry for the very long message
If you're interested, I found completely by chance a wikipedia version of what I did : Mellin transform - Wikipedia, the free encyclopedia