Asymptotic formulas for generalized elliptic-type integrals
In this paper the authors consider a generalized family of elliptic-type integrals in the following form: (*) \(\Lambda^{(\alpha, \beta)}_{(\lambda, \gamma,\mu)} (\rho,\delta;k) = \int^\pi_0 \cos^{2\alpha-1} (\theta/2) \sin^{2\beta-1} (\theta/2) (1-k^2 \cos \theta)^{-\mu -1/2} \cdot [1-\rho \sin^2 (\theta/2)]^{-\lambda} \cdot [1+\delta \cos^2 (\theta/2)]^{- \gamma}d \theta\), \(0\leq k<1\); \(\text{Re} \alpha, \text{Re} \beta>0\); \(\lambda,\mu, \gamma\in\mathbb{C}\); either \(|\rho |\), \(|\delta |<1\) or \(\rho\), (or \(\delta) \in\mathbb{C}\), whenever \(\lambda= -m\) (or \(\gamma= -m)\), \(m\in\mathbb{N}_0\). They first express (*) in terms of the Lauricella's hypergeometric function of three variables \(F^{(3)}_D\); next, they obtain its asymptotic expansion as \(k^2\to 1\). In particular, \(\Lambda^{(\alpha, \beta)}_{(\lambda, \gamma,\mu)} (\rho,0;k)= \Lambda^{(\alpha, \beta)}_{(\lambda, 0,\mu)} (\rho,\delta;k) =\Lambda^{(\alpha, \beta)}_{\lambda, \mu} (\rho;k)\), which has been studied by \textit{H. M. Srivastava} and \textit{R. N. Siddiqi} [Radiat. Phys. Chem. 46, 303-315 (1995)]. By choosing suitable parameters the corresponding asymptotic formulas for other special cases of elliptic-type integrals, such as \(R_\mu (k,\alpha, \gamma)\) [see \textit{M. L. Glasser} and \textit{S. L. Kalla}, Rev. Téc. Ing., Univ. Zulia 12, No. 1, 47-50 (1989; Zbl 0681.33002)] and \(\Omega_j(k)\) [see \textit{L. F. Epstein} and \textit{J. H. Hubbell}, J. Res. Natl. Bur. Stand., Sect. B 67, 1-17 (1963; Zbl 0114.64002)], are also derived.
scientific article; zbMATH DE number 950649
scientific article