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I have this two integral equations which are an Hankel transform pair. I gotta combine them to find $A(k)$

$$ R^{n+{1}/{4}} \, \Sigma(R,t=0) \, = \, \int_0^\infty [A(k)\, k^{-1}] \, J_l(ky) \, k \, dk $$

and

$$ A(k)\, k^{-1} \, = \, \int_0^\infty [R^{n+{1}/{4}} \, \Sigma(R,t=0)]\, J_l(ky) \, y \, dy $$

My text says: combining them we obtain

$$ A(k) \, = \, \Bigl(1- \frac{n}{2}\Bigr)^{-1} \int_0^\infty \Sigma(y',0) J_l(ky') \, k \, R'^{\frac{5}{4}} dR' $$

Now I'd like to ask you how to reach this result combining them. I don't even get if the apex means derivation or it's just formal symbol used to point out the difference between $y'$ and $R'$. I don't even get how the integration switches from those variables to $R'$ in the last integral.

Thank you in advance for any kind of help.

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  • $\begingroup$ Welcome to math.SE! I removed the combinations tag. When in doubt, please avail yourself of the tag summaries in choosing tags. $\endgroup$ – joriki Jul 5 '16 at 10:43
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    $\begingroup$ Ok, sorry for that, I read it but since english is not my first language i misunderstood it. $\endgroup$ – Run like hell Jul 5 '16 at 10:58

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