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given two functios $ f(x) $ and $ g(x) $ related by $\frac{ \Gamma(s-1/2)}{\Gamma(s) \sqrt{ \pi}}\int_{0}^{\infty}dx \frac{g(x)dx}{(x+y)^{s-1/2}}=\int_{0}^{\infty}dx \frac{f(x)dx}{(x+y)^{s}}$ what relation exists between them ? I believe that
$ g(x)= A \frac{d^{1/2}f(x)}{dx^{1/2}}$ for some constant $A$ but I am not sure.

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    @Jose I think it can be checked using Laplace transform, the formula http://en.wikipedia.org/wiki/Laplace_transform#Properties_and_theorems for cross-correlation http://en.wikipedia.org/wiki/Cross-correlation and the fact that Laplace transform turns fractional derivatives into multiplication on power function.2012-07-07

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