Expressing an integral in terms of gamma functions

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CAF123
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I want to show that $$\int_0^{\infty} \frac{ds}{s-q^2} \frac{s^{-1-\epsilon}}{s-t \frac{z}{1-z}} = \Gamma(1-\epsilon) \Gamma(\epsilon) \frac{1}{t \frac{z}{1-z} - q^2} \left((-t)^{-1-\epsilon} \left(\frac{z}{1-z}\right)^{-1-\epsilon} -(-q^2)^{-1-\epsilon}\right) $$

I have many ideas on how to obtain the final answer but I haven't quite got there. I can rexpress the two denominator terms in terms of partial fractions to give $$ \frac{1}{q^2 - t\frac{z}{1-z}}\int_0^{\infty} ds \,s^{-1-\epsilon} \left( \frac{1}{s-q^2} - \frac{1}{s - t\frac{z}{1-z}}\right)$$

Consider the first integral: $$\int_0^{\infty} ds \, \frac{s^{-1-\epsilon}}{s-q^2}.$$ If I perform integration by parts I get $$\int_0^{\infty} ds \, \frac{s^{-1-\epsilon}}{s-q^2} = s^{-1-\epsilon}\ln|s-q^2| |^{\infty}_0 + (1+\epsilon) \int_0^{\infty}ds\, s^{-2-\epsilon} \ln |s-q^2|$$ I then thought about using the fact that $$\ln w = \frac{d}{d\alpha} w^{\alpha}|_{\alpha=0}$$ to have everything in terms of rationals but the subsequent integral is no more tractable as far as I can tell. I also thought about using 'Schwinger parameters' but again not quite of the form required.

Then I went back to the starting integral and thought about feynman parametrisation https://en.wikipedia.org/wiki/Feynman_parametrization to rexpress the two denominators in terms of an integral but again, the resulting integral did not lead anywhere.

Any hints or suggestions would be great! Thanks
 

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Thanks for the post! This is an automated courtesy bump. Sorry you aren't generating responses at the moment. Do you have any further information, come to any new conclusions or is it possible to reword the post?
 
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haruspex
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I tried working backwards from the form ##\Gamma(1-\epsilon)\Gamma(\epsilon)##.
I wrote that as a product of two integrals, then merged them into a double integral.
A change of coordinates produced a different double integral which could be factored back out to two single integrals again.
One of them was straightforward, the other produced ##\int \frac{y^{\epsilon-1}(1+y)}{1+y^2}dy##. Not sure whether that can be got into the form you need, or maybe I made a mistake somewhere. Looks close though.
 

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