Hello,

I'm trying to solve the following integral :

$\int_0^\infty \frac{1}{t^{d/2}}(e^{-\gamma t} - e^{-\delta t})dt$.

I know it equals

$\Gamma(1-\frac{d}{2})[\gamma^{\frac{d}{2}-1}-\delta^{\frac{d}{2}-1}]$ for every $d<4$.

However, this does not work for $d=2$ as the gamma function is not defined in zero. According to some reference (E. Akkermans and G. Montambaux, Mesoscopic Physics of Electrons and Photons, ISBN 978-0521855129 (2007)), it is possible to solve it and the resulting law they obtain is logarithmic. However, they do not give any details in between.

Any advice on how to proceed? Which integration method would work here?

For the story, this integral is part of the cooperon correction to the conductivity which causes weak localization in mesoscopic systems.

Thanks in advance!