If it exists, what is the residue of $e^x x^{-r-1} \log x$ at $x=0$?
Thanks in advance for any kind of help!
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2$x=0$ is not an isolated singularity in this case, so we can't talk about residue. – C. Dubussy Aug 02 '16 at 15:58
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Thanks! What I was trying to do was to calculate the Cauchy integral $\oint_{|x|=1} e^x x^{-r-1} \log x dx$. I was asking myself whether that could be calculated using the Residue theorem. But as I see, the answer is no. – Matthias Aug 02 '16 at 16:09
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for assigning a residue to a branch point, you have to consider something like a Bromwich contour – reuns Aug 02 '16 at 16:50
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@Matthias of course you can't use the residue theorem, $e^x x^{-r-1} \log x$ is not holomorphic on $|x| = 1$ (it is not even continuous) – reuns Aug 02 '16 at 16:51
