On Infinite Integrals Containing Products of Bessel Functions

Abstract

where the only conditions laid upon the parameters are those necessary to secure convergence, is indeed a difficult one for N greater than two. Some special cases have been evaluated for the case N equal to three, but Watson [1, p. 16] has conjectured that a general solution is impossible in this case. For N finite and equal to four or more, the only solutions known to the author are those due to Bailey [2, p. 47]; these make the value of n dependent on the mA's, and require that the Pr's be of such values as to be incapable of being formed into a polygon. While the value of a general closed form solution can not be questioned, nevertheless, the applied mathematician often has need of approximate results which are of sufficient accuracy to be useful. It is the purpose of this paper to present an approximate method by which (1) may be evaluated, with the only restrictions being those necessary for convergence, together with the arbitrary assumptions that the m,'s, P,'s, and n are all positive. This not only simplifies the analysis, but the method then has direct application to the approximate calculation of the modulation products resulting from the application of N sinusoids of incommensurable frequency to an nth law unbiased rectifier. The method is perfectly general and may probably be applied to other integrals related to (1). The method takes advantage of the generally rapid convergence of the integrand in (1), caused by the x'+' term in the denominator. To do this, an algebraic-exponential power series is derived which is a good fit for the Bessel function in the region of small argument. The first term in this series is the so-called Laplace approximation to the Bessel function, which is commonly used in analyzing random noise in non-linear devices where the noise is considered to be composed of a very large number of sinusoidal waves. We begin by writing down the absolutely convergent power series for the Bessel function