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.
integral equation solution for two functions $ f(x) $ and $ g(x) $ and see if they are related
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integration
integral-equations
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2Could you please explain the notation $\int_0^\infty dx F(x)dx$. – 2012-07-06
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0Also: there is a $y$ on the right hand side and none on the left hand side. Should the relation hold for all $y$? – 2012-07-06
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0@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