Bessel integrals and sums: Old and new.

摘要

Bessel functions are among the most important special functions, having diverse applications to physics, Engineering, and Mathematical analysis itself. In particular, the integral representations for products of Bessel functions are useful for evaluating definite integrals that contain products of two Bessel functions under the integral sign. In addition, they may be considered as a natural generalizations of a well-known trigonometric identities. In the first part of this thesis, we prove some known integral representations for the product of two Bessel functions of the same argument and same order, one of which is Nicholson's formula. Then we prove some new integral representations for the product of two Bessel functions of different arguments and different orders and a combination of their products, which generalizes Nicholson's integral and related formulas.Sonine-Gegenbauer type integrals have been studied since 1880. These integrals are important in some physical applications. Some of these integrals have been evaluated in terms of hypergeometric functions of two and three variables. In the second part of this thesis, we evaluate some special cases of these integrals in terms of familiar functions. In our computation of these integrals and some related integrals we utilize Parseval's formula. In this part of this thesis we also investigate a special class of Bessel integrals where we summarize our results in a theorem regarding this class of integrals.In the third part of this thesis, we investigate various new Neumann series which are of special interest in mathematical analysis. In deriving such series we rely on a special case of the Poisson summation formula. As a consequence of our results we obtain the sum of some interesting trigonometric series.In the last part of this thesis, we summarize our results and outline directions for future research.