# contour integral with singularities on the contour

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• Nov 28th 2012, 11:16 AM
tenderline
contour integral with singularities on the contour
I want to compute the following integral
$\displaystyle \oint_{|z|=1}\frac{\exp(\frac{1}{z})}{z^2-1}\,dz$
The integrand has essential singularity at the origin, and 2-poles at $\displaystyle \pm 1$,which lie on the curve $\displaystyle |z|=1$ so i can't apply residue formula. How can i proceed?
• Nov 28th 2012, 04:06 PM
SworD
Re: contour integral with singularities on the contour
Ive never seen integration where the singularities lie on the contour, is that even defined?
• Nov 28th 2012, 04:27 PM
coolge
Re: contour integral with singularities on the contour
You need to define the contour with small semi-circles around the poles of radius some \delta and then take the limit as \delta tends to zero. In that way the function will be analytic in the domain that has avoided the singularities. When f is analytic inside a domain D the integral over D of f = 0. Check any book on complex analysis. You can find the details.