On the Epstein-Hubbell generalized elliptic-type integral
Many authors, e.g. L. F. Epstein, J. H. Hubbell, J. D. Evans and V. D. Evans, dealt with the power series expansion of the Epstein-Hubbell generalized elliptic-type integral \[ \Omega_ j (K):= \int^ \pi_ 0 \bigl( 1-K^ 2 \cos (\theta) \bigr)^{-j-1/2} d \theta,\quad 0 \leq K<1,\;j = 0,1,2, \dots . \] In the paper being reviewed, the authors P. G. Todorov and J. H. Hubbell present an elegant proof of the well known power series expansion of \(\Omega_ j\). The proof is based on the transformation of the integral via \(z = e^{i \theta}\) into a complex integral, the insertion of the Laurent series expansion of the integrand and the change of the order of integration and summation. Furthermore, it is shown by a straightforward calculation that the power series expansion of \(\Omega_ j\) can be written by means of the Gauss hypergeometric function, i.e. \[ \Omega_ j(K) = \pi F \left( {1 \over 4} + {j \over 2}, {3 \over 4} + {j \over 2}; 1; K^ 4 \right), \quad 0 \leq K<1,\quad j = 0,1,2,\dots. \]
scientific article; zbMATH DE number 566124
scientific article