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Can this integral be done analytically? \begin{eqnarray} & & \int\limits^1_0 d\eta \: \eta^{ 1 - 2 \epsilon}\, ( 1 - \eta)^{1/2 - \epsilon} \, \left( 1 + 2 \sqrt{z} - \left( 1 - 2 \sqrt{z} \right) \eta \right)^{1/2 - \epsilon} \left(z + \left( 1 - 2 \sqrt{z} \right) \eta \right)^{-2 + \epsilon} \\ && \qquad {} \times {}_2 F_1 \left(1, \epsilon ; 2 - \epsilon; \frac{ z }{ z + \left( 1 - 2 \sqrt{z} \right) \eta }\right) \end{eqnarray} with $0 < z < \frac{1}{4}$.

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kld, since you are a new user, here are a few things about the site you should know: 1. To get the best possible answers, it is helpful if you say where the problem originated, 2. You will get a better response, if you indicate, what you have already tried to answer the question yourself. 3. If this is homework, please add the [homework] tag; people will still help, so don't worry. and finally: Welcome to math.SE! – draks ... Jul 19 '12 at 11:55
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@draks, somehow the hypergeometric function being there makes me doubt that this is homework... – J. M. Jul 19 '12 at 11:56
@J.M. , depends. Didn't you do any homework at university? Sure you did... – draks ... Jul 19 '12 at 12:03
@draks, well, all the things I know about special functions, I never learned at the university... :) – J. M. Jul 19 '12 at 12:05
I believe this kind of integrals comes from statistical and physical applications. – Mhenni Benghorbal Jul 19 '12 at 13:25

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