Abstract
The main object of this article is to present a systematic study of integral representations for generalized Mathieu series and its alternating variant, and to derive a new integral expression for these special functions by contour integration using rectangular integration path. Also, by virtue of newly established integral form of generalized Mathieu series, we obtain a new integral expression for the Bessel function of the first kind of half integer order, solving a related Fredholm integral equation of the first kind with nondegenerate kernel.
Highlights
AND PRELIMINARIES The series (1.1) S(r) = (n2 2n + r2 )2, n≥1 r ∈ R+, was introduced and studied by Emile Leonard Mathieu (1835-1890) in his book [9] devoted to the elasticity of solid bodies
N≥1 r ∈ R+, was introduced and studied by Emile Leonard Mathieu (1835-1890) in his book [9] devoted to the elasticity of solid bodies
Which was introduced by Pogany et al in [14]
Summary
The main object of this article is to present a systematic study of integral representations for generalized Mathieu series and its alternating variant, and to derive a new integral expression for these special functions by contour integration using rectangular integration path. By virtue of newly established integral form of generalized Mathieu series, we obtain a new integral expression for the Bessel function of the first kind of half integer order, solving a related Fredholm integral equation of the first kind with nondegenerate kernel
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