Integrals of some three Bessel functions and Legendre functions. II

Integrals of three Bessel functions of the form ∫∞0Jμ(at)Jν(bt) ×H(1)ρ(ct)dt are calculated when μ,ν,ρ,a,b, and c are arbitrary real numbers. For this, use is made of the factorization of the Appell function F4 in two hypergeometric functions. Further simplifications occur if μ=±ν or ρ=±1/2. New results are given, mainly when real a, b, and c satisfy the inequalities ‖a−b‖<c<a+b, which correspond to most physical situations.

计算当\(\mu\)、\(\nu\)、\(\rho\)、\(a\)、\(b\)和\(c\)为任意实数时，形如\(\int_{0}^{\infty}J_{\mu}(at)J_{\nu}(bt)H_{\rho}^{(1)}(ct)dt\)的三个贝塞尔函数的积分。为此，利用阿佩尔函数\(F_4\)分解为两个超几何函数。如果\(\mu = \pm\nu\)或\(\rho = \pm\frac{1}{2}\)，会进一步简化。给出了新的结果，主要是当实数\(a\)、\(b\)和\(c\)满足不等式\(\vert a - b\vert < c < a + b\)时，这对应于大多数物理情形。