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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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Could you please explain the notation $\int_0^\infty dx F(x)dx$. –  Mercy Jul 6 '12 at 18:55
Also: there is a $y$ on the right hand side and none on the left hand side. Should the relation hold for all $y$? –  Fabian Jul 6 '12 at 18:57
@Jose I think it can be checked using Laplace transform, the formula en.wikipedia.org/wiki/Laplace_transform#Properties_and_theorems for cross-correlation en.wikipedia.org/wiki/Cross-correlation and the fact that Laplace transform turns fractional derivatives into multiplication on power function. –  Andrew Jul 7 '12 at 6:40

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