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Integral operator

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One form of the Abel transform is
$$
g(y)=\int_0^y\frac{f(x)}{\sqrt{y^2-x^2}}\,dx.
$$
For sufficiently regular $f$, it is inverted by
$$
f(x)=\frac2\pi\frac d{dx}\int_0^x
\frac{y g(y)}{\sqrt{x^2-y^2}}\,dy.
$$
Endpoint atoms must be handled separately as measures rather than ordinary densities.


Abel transform

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Abel transform

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Abel transform

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