We study the Abel differential equations that admits either a generalized Weierstrass first integral or a generalized Weierstrass inverse integrating factor.

Legendre coefficients of an integrable function f(x) are proved to coincide with the Fourier a nonnegative index suitable Abel-type transform itself. The numerical computation N can thus be carried out efficiently in O(NlogN) operations by means single fast f(x). Symmetries associated exploited further reduce computational complexity. dual problem calculating sum expansions at prescribed set po...

Abstract In the case of radiography of a cylindrically symmetric object, the Abel transform is useful for describing the tomographic measurement operator. The inverse of this operator is unbounded, so regularization is required for the computation of satisfactory inversions. We introduce the use of the total variation seminorm for this purpose, and prove the existence and uniqueness of solution...

By means of series–rearrangements and finite differences, elementary proofs are presented for the well–known convolution identities of Abel and Hagen–Rothe.