Treffer: On a Class of Generalized Elliptic-type Integrals: On a class of generalized elliptic-type integrals

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. The authors consider the new class of generalized elliptic-type integrals, of the forms \[ A^{(\alpha,\beta)}_{(a,b,c)} (k) = A^{(\alpha,\beta)}_{(a_1,\dots,a_n,b_1,\dots,b_n,c_1,\dots,c_n)} (k) \] \[ = \int_0^{\pi} \cos^{2\alpha-1} \biggl(\frac {\theta}2\biggr) \sin^{2\beta-1}\biggl(\frac {\theta}2\biggr) \prod_{j=1}^n \left\{ {}_2F_1 \left(a_j,b_j,c_j; \biggl({\frac {k_j^2}{k_j^2-1}}\biggr) (1-\cos \theta)\right)\right\} d\theta \tag{*} \] \[ = \int\limits_0^1 (1-u)^{\alpha-1} u^{\beta-1} \prod_{j=1}^n {}_2F_1 \left[a_j,b_j,c_j; {\frac {2k_j^2 u} {k_j^2-1}}\right]du, \] \[ \Re(\alpha) >0, \quad \Re(\beta)>0,\quad |k_j