<s>[INST] How to calculate $\lim_{x\to 1}\int_{0}^{1}\frac{dy}{\sqrt{1-y^{2}}}\frac{y^{3/2}}{\sqrt{x - y}}$ when $x>1$? Numerically, it looks that the limit is 
$$\lim_{x\to 1}\int_{0}^{1}\frac{dy}{\sqrt{1-y^{2}}}\frac{y^{3/2}}{\sqrt{x - y}} = \frac{1}{\sqrt{2}}\log(1 - x) + cte $$,
but I have not been able to demonstrate it analytically. Does anyone have a idea on how to deal with this limit?
 [/INST] I have found how to handle with this problem. The key point is to use the following integral representation of $\frac{1}{\sqrt{x-y}}$:
$\frac{1}{\sqrt{x-y}} = \frac{1}{\sqrt{\pi}}\int_{0}^{\infty}\frac{da}{\sqrt{\pi}} e^{-a(x-y)}$.
Using Mathematica, we can carry out the integral over $y$ and $a$ (in this order), obtaining a complex result in terms of generalized hypergeometric functions
$\int_{0}^{1}\frac{dy}{\sqrt{1-y^{2}}}\frac{y^{3/2}}{\sqrt{x - y}} = 
\frac{8 \sqrt{\frac{2}{\pi }} \left(2 x \Gamma \left(\frac{5}{4}\right) \Gamma \left(\frac{9}{4}\right) \, _3F_2\left(\frac{1}{4},\frac{3}{4},\frac{5}{4};\frac{1}{2},\frac{7}{4};\frac{1}{x^2}\right)+\Gamma \left(\frac{7}{4}\right)^2 \, _3F_2\left(\frac{3}{4},\frac{5}{4},\frac{7}{4};\frac{3}{2},\frac{9}{4};\frac{1}{x^2}\right)\right)}{15 x^{3/2}}.$
After that, the limit is easily obtained by Mathematica as
$\lim_{x\to 1}\int_{0}^{1}\frac{dy}{\sqrt{1-y^{2}}}\frac{y^{3/2}}{\sqrt{x - y}} = \frac{1}{\sqrt{2}}\log(1-x) + \frac{75 \log (2)-4 \left(5 \, _3F_2\left(\frac{1}{4},1,\frac{3}{2};\frac{5}{4},\frac{7}{4};1\right)+\, _3F_2\left(\frac{3}{4},1,\frac{3}{2};\frac{7}{4},\frac{9}{4};1\right)\right)}{15 \sqrt{2}}$,
which is the expected result.
Thanks by your help.
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