The following integral transform relationship, known as the Abel transform, exists between two functions f(x) and g(t) for 0<α<1, f(x) | = | integral_0^x (g(t) d t)/(x - t)^α g(t) | = | (sin(πα))/π d/(d t) integral_0^t (f(x) d x)/(t - x)^(1 - α) | = | (sin(πα))/π[ integral_0^t (d f)/(d x) (d x)/(t - x)^(1 - α) + (f(0))/t^(1 - α)]. The Abel transform is used in calculating the radial mass distribution of galaxies and inverting planetary radio occultation data to obtain atmospheric information as a function of height. Bracewell defines a slightly different form of the Abel transform given by g(x) = A[f(r)] = 2 integral_x^∞ (f(r) r d r)/sqrt(r^2 - x^2).

Fourier transform | Hilbert transform | integral equation