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Abel Transform

Niels Abel.

Abel transform, named after Niels Abel, is an integral transform often used in the analysis of spherically symmetric or axially symmetric functions.

The following integral transform relationship exists between two functions f(x)f\left(x\right) and g(t)g\left(t\right) for 0<α<10 < \alpha < 1,

f(x)=∫0xg(t)(x−t)αdt\displaystyle f \left( x \right) = \int_0^x \frac{g \left( t \right)}{\left( x-t \right)^\alpha} \mathrm{d}t
g(t)=sin⁡(πα)πddt∫0tf(x)(t−x)1−αdx=sin⁡(πα)π[∫0tdfdx1(t−x)1−αdx+f(0)t1−α]\begin{aligned} g \left( t \right) &= \frac{\sin \left(\pi \alpha \right)}{\pi} \frac{\mathrm{d}}{\mathrm{d}t} \int_0^t \frac{f \left( x \right)}{\left( t-x \right)^{1- \alpha}} \mathrm{d}x \\ &= \frac{\sin \left(\pi \alpha \right)}{\pi} \left[ \int_0^t \frac{\mathrm{d}f}{\mathrm{d}x} \frac{1}{\left( t-x \right)^{1- \alpha}} \mathrm{d}x + \frac{f \left( 0 \right)}{t^{1- \alpha}} \right] \end{aligned}

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