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Integrals of some three Bessel functions and Legendre functions. II. (English) Zbl 0586.33005

Integrals of three Bessel functions of the form

${\int }_{0}^{\infty }{J}_{\mu }\left(at\right){J}_{\nu }\left(bt\right){H}_{\rho }^{\left(1\right)}\left(ct\right)dt$

are calculated when $\mu$,$\nu$,$\rho$,a,b, and c are arbitrary real numbers. For this, use is made of the factorization of the Appell function ${F}_{4}$ in two hypergeometric functions. Further simplifications occur if $\mu =±\nu$ or $\rho =±1/2$. New resuls are given, mainly when real a,b, and c satisfy the inequalities $|a-b|, which correspond to most physical situations.

##### MSC:
 33C10 Bessel and Airy functions, cylinder functions, ${}_{0}{F}_{1}$ 33C55 Spherical harmonics