- #1
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Homework Statement
I have never formally studied complex analysis, but I am reading this paper: http://adsabs.harvard.edu/abs/1996MNRAS.283..837S
wherein section 2.2 they make use of the residue theorem. I am trying to follow along with this (and have looked up contour integration, cauchy's formula etc..).
I want to compute the integral
##I_1 = \int_{0}^{2\pi} \frac{d \phi}{(X-Y)^2}## where X and Y are complex variables
Homework Equations
The Attempt at a Solution
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Using the substitution ##X = |\vec x|e^{i\phi}##, ##dX = i X d\phi## we can write this as
##I_1 = -\oint \frac{i dX}{X(X-Y)^2}## which has two poles, at X=0 and X=Y.
The residue theorem allows me to say that the result of this integral is ##2 \pi i## times the sum of the residues. I have been able to compute the residue by expanding at ##X=0## as follows:
##f(X) = - \frac{1}{X(X-Y)^2} = - \frac{1}{X}(\frac{1}{(X-Y)^2})##
I then taylor expand ##\frac{1}{(X-Y)^2}## around X=0 and find:
##f(X) = -\frac{1}{X}(\frac{1}{Y^2} + \frac{2X}{Y^3} + ...)##
and so by identifying the residue with the term proportional to ##\frac{1}{X}##, I see that the residue of the function expanded at that pole is ##- \frac{1}{Y^2}##
I am stuck with how to find the residue of the ##X=Y## expansion as it can't be done in the same way as above.
Thank you for any hints you can give!
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