On a class of generalized elliptic-type integrals (Q2761693)

The elliptic-type integrals, besides presenting an old branch of the classical analysis, have found recently rather frequent usage in problems of radiation physics, polution, engineering and other applied sciences. These applications have stimulated many authors to consider various families of elliptic-type integrals generalizing the well known complete elliptc integrals \(K(k)\) and \(E(k)\), by involving wide classes of hypergeometric functions as kernels, depending on sets of parameters. NEWLINENEWLINENEWLINEThe authors consider the new class of generalized elliptic-type integrals, of the forms NEWLINE\[NEWLINE A^{(\alpha,\beta)}_{(a,b,c)} (k) = A^{(\alpha,\beta)}_{(a_1,\dots,a_n,b_1,\dots,b_n,c_1,\dots,c_n)} (k) NEWLINE\]NEWLINE NEWLINE\[NEWLINE = \int_0^{\pi} \cos^{2\alpha-1} \biggl(\frac {\theta}2\biggr) \sin^{2\beta-1}\biggl(\frac {\theta}2\biggr) \prod_{j=1}^n \left\{ {}_2F_1 \left(a_j,b_j,c_j; \biggl({\frac {k_j^2}{k_j^2-1}}\biggr) (1-\cos \theta)\right)\right\} d\theta \tag{*} NEWLINE\]NEWLINE NEWLINE\[NEWLINE = \int\limits_0^1 (1-u)^{\alpha-1} u^{\beta-1} \prod_{j=1}^n {}_2F_1 \left[a_j,b_j,c_j; {\frac {2k_j^2 u} {k_j^2-1}}\right]du,NEWLINE\]NEWLINE NEWLINE\[NEWLINE \Re(\alpha) >0, \quad \Re(\beta)>0,\quad |k_j|<1, \quad j=1,\dots,n. NEWLINE\]NEWLINE This paper should be considered as a continuation of the same authors' work, where recurrence relations and asymptotic formulas for these generalized elliptic-type integrals have been proposed, [\textit{M. Garg, V. Katta} and \textit{S. L. Kalla}, Appl. Math. Comput. (2001; to appear)]. NEWLINENEWLINENEWLINEHere, the following new results for these elliptic-type integrals are given: single and multiple integral formulas involving (*) and giving some Kampé de Feriet functions and generalized Lauricella functions of several variables; differentiation and fractional integral formula (with respect to \(k\)); some single-terms approximation formulas, as useful tools for the applied scientsts. NEWLINENEWLINENEWLINEIt is commented that studying the class (*) incorporates, as very special cases some classical studies as by \textit{L. F. Epstein} and \textit{J. H. Hubbell} [J. Res. Natl. Bur. Stand., Sect B 67, 1-17 (1963; Zbl 0114.06402)], \textit{S. L. Kalla, S. Conde} and \textit{J. H. Hubbell} [Appl. Anal. 22, 273-287 (1986; Zbl 0597.33005)], as well as some more recent generalizations, as by \textit{S. L. Kalla} and \textit{V. K. Tuan} [Comput. Math. Appl. 32, 49-55 (1996; Zbl 0891.33010))], \textit{A. Al-Zamel, V. K. Tuan} and \textit{S. L. Kalla} [Appl. Math. Comput. 114, 13-25 (2000; Zbl 1049.33017], etc.