Numerical computation of incomplete Lipschitz-Hankel integral \(Je_ 0(a,z)\)
Incomplete Lipschitz-Hankel integrals are important special functions since they arise in a number of problems in mathematical physics. The authors consider especially the integral \(\int^{z}_{0}e^{-at}J_ 0(t)dt\), \(J_ 0\) the Bessel function. Convergent and asymptotic factorial-Neumann series expansions are used in an algorithm for computing the integral to a user defined number of significant digits. [A ``factorial-Neumann series'' can be written in the form \(\Sigma a_ nz^ nJ_{n+r}(z)\). \textit{N. Nielsen} in the classical ``Handbuch der Theorie der Cylinderfunktionen'', (1904; JFM 35.0476.03) classified this series as ``Fakultätenreihe'', different from the usual terminology, due to E. Landau a.o..] The algorithm offers a very accurate and efficient way to compute the integral, particularly when compared with an adaptive quadrature routine. The algorithm accomplishes this by choosing automatically which of the asymptotic expansions to use.
asymptotic factorial-Neumann series expansions
scientific article; zbMATH DE number 4149441