Infinite integrals involving Bessel functions by an improved approach of contour integration and the residue theorem

Abstract

Two lemmas are established to evaluate integrals involving Bessel functions or products of two Bessel functions on a large semicircle. These lemmas are as useful as Jordan’s lemma. With the help of them, the approach of contour integration and residue theorem to infinite integrals becomes much more powerful than ever before, yet is still simple and easy. A general formula is derived for infinite integrals involving rational functions, powers and Bessel functions, the index of the power and the order of the Bessel function being independent, and the result is expressed in terms of residues and coefficients of the Laurent series of the rational function. Similar formulae are presented for infinite integrals involving Neumann or Struve functions. Another general formula is derived for infinite integrals involving rational functions, powers and two Bessel functions, the index of the power and the orders of the Bessel functions being independent. The formulation presented may be extended to infinite integrals of similar types. Typical examples are worked out for each type of integral, and many of the results obtained seem not available in the literature.