Gauss-Legendre approximations for the Hubbell rectangular-source integral
Gauss-Legendre quadrature is used to generate some new approximation formulas for \(f(a,b)\), where \(f(a,b)\) is the Hubbell rectangular-source integral, \(0 < a \leq b <\infty\). Numerical values of \(f(a,b)\) are used in radiation physics applications. This function is closely related with Appell's hypergeometric function. The author presents a family of fairly efficient nonlinear approximations \(f(a,b) \approx \Omega_n (a,b)\), \(n = 1,2,\dots,\) that can be used to compute \(f(a,b)\) to a prescribed relative error, by taking \(n\) sufficiently large. The minimum number of significant decimal digits of accuracy attained by a particular approximation \(f(a,b) \approx \Omega_n (a,b)\), over the triangular domain \(0.1 \leq a \leq b \leq 20\), is ascertained from a table provided for the cases \(n = 1(1)4\) and \(6(2)16\).
scientific article; zbMATH DE number 770361