Abel Transform

The following Integral Transform relationship, known as the Abel transform, exists between two functions $f(x)$ and $g(t)$ for $0<\alpha<1$,

$$f(x) = \int_0^x {g(t)\,dt\over(x-t)^\alpha}$$ (1)

$$g(t) = -{\sin(\pi\alpha)\over\pi} {d\over dt}\int_0^t {f(x)\,dx\over(x-t)^{1-\alpha}}$$ (2)

$$= -{\sin(\pi\alpha)\over\pi}\left[{\int_0^t{df\over dx} {dx\over(t-x)^{1-\alpha}} + {f(0)\over t^{1-\alpha}}}\right].$$ (3)

The Abel transform is used in calculating the radial mass distribution of galaxies and inverting planetary radio occultation data to obtain atmospheric information.

References

Arfken, G. Mathematical Methods for Physicists, 3rd ed. Orlando, FL: Academic Press, pp. 875-876, 1985.

Binney, J. and Tremaine, S. Galactic Dynamics. Princeton, NJ: Princeton University Press, p. 651, 1987.

Bracewell, R. The Fourier Transform and Its Applications. New York: McGraw-Hill, pp. 262-266, 1965.