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let be the integral equation for two functions $ f(x) $ and $ g(x) $

in the form $$ g(s)= \int_{0}^{s}\sqrt{s-f(x)}dx $$

is valid to accept that in the sense of fractional calculus, the ONLY solution to this equation will be

$$ f^{-1}(x)= a \frac{d^{1/2}}{dx^{1/2}}g(x) $$

here of course $ f^{-1}(x) $ means the inverse function of $ f(x) $ so $ f(f^{-1}(x))=x $

for some real constant 'a'

EDIT: sorry i forgot in order to obtain the solution i make the change of variables $ x=f^{-1}(t) $ inside the first nonlinear equation.

my interest is due to the fact that the two integral equations

$ g(s)= \int_{0}^{s}dt \frac{f(t)}{\sqrt{s-t}}$ and $ g(s)= \int_{0}^{s}\sqrt{s-f(t)}dt $ appear in several branches of physics dealing with WKB approximation.

i say so because this is important for my theoretical model implying the RIemann zeros :) thanks in advance

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What is the question here? I don't see one. –  Tim Seguine Jan 16 '13 at 14:57
    
i want to solve a nonlinear integral equation, i am asking if my solution proposed for $ f^{-1}(x) $ is correct –  Jose Garcia Jan 16 '13 at 18:12
    
Ok, I guessed that this was probably what you were driving at, but the fact remains that there is nothing in your post that is grammatically a question. –  Tim Seguine Jan 17 '13 at 18:05

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