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How to calculate the contour integration with branch point?

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The question come from a Mutusbara Sum like this $${ \sum _{ { z=i\omega }_{ n } } { \frac { -\alpha E\pi }{ 4{ z }^{ 3 }\sqrt { -\alpha -z } } } }$$ it equal a contour integral around Imaginary axis with pole($\omega_n=\frac{(2n+1)\pi}{\beta}$,Fermion) $${ { \frac { 1 }{ 2\pi i } \oint { \frac { -\beta }{ { e }^{ \beta z }+1 } \quad \frac { -\alpha E\pi }{ 4{ z }^{ 3 }\sqrt { -\alpha -z } } } } }-{ Res[\frac { -\beta }{ { e }^{ \beta z }+1 } \quad \frac { -\alpha E\pi }{ 4{ z }^{ 3 }\sqrt { -\alpha -z } } ] }_{ z=0 }$$ the 1st integral have Branch point at $-\alpha$($\alpha \in Reals$),when I inflate original imaginary path to Infinite diameter, only the path around the pole $z=-\alpha$ and two parallel(but opposite direction) paths survive. Further more, I reverse the direction of these paths, and the 1st integral convert to $$-{ { \frac { 1 }{ 2\pi i } \oint { \frac { -\beta }{ { e }^{ \beta z }+1 } \quad \frac { -\alpha E\pi }{ 4{ z }^{ 3 }\sqrt { -\alpha -z } } } } }$$ on a path around the branch point $z=-\alpha$. My question is How to calsulate this contour integral? If any one give some help or hint, I would appreciate!!

This post imported from StackExchange Physics at 2016-06-17 12:17 (UTC), posted by SE-user alxandernashzhang
asked May 16, 2016 in Mathematics by alxandernashzhang (5 points) [ no revision ]
retagged Jun 17, 2016

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