arXiv subjects
Publications and source records attributed to Diego E. Dominici.
We consider a new identity involving integrals and sums of Bessel functions. The identity provides new ways to evaluate integrals of products of two Bessel functions. The identity is remarkably simple and powerful since the summand and integrand are of exactly the same form and the sum converges to the integral relatively fast for most cases. A proof and numerical examples of the identity are discussed.
In this paper we generalize an identity first given by Heinrich Eduard Heine in his treatise, {\it Handbuch der Kugelfunctionen, Theorie und Anwendungen (1881), which gives a Fourier series for $1/[z-\cosψ]^{1/2}$, for $z,ψ\in\R$, and $z>1$, in terms of associated Legendre functions of the second kind with odd-half-integer degree and vanishing order. In this paper we give a generalization of this identity as a Fourier series of $1/[z-\cosψ]^μ$, where $z,μ\in\C$, $|z|>1$, and the coefficients of the expansion are given in terms of the same functions with order given by $\frac12-μ$. We are also able to compute certain closed-form expressions for associated Legendre functions of the second kind.