Integral representations and summations of the modified Struve function

Abstract

It is known that Struve function Hν and modified Struve function Lν are closely connected to Bessel function of the first kind Jν and to modified Bessel function of the first kind Iν and possess representations through higher transcendental functions like generalized hypergeometric 1 F 2 and Meijer G function. Also, the NIST project and Wolfram formula collection contain a set of Kapteyn type series expansions for Lν (x). In this paper firstly, we obtain various another type integral representation formulae for Lν (x) using the technique developed by D. Jankov and the authors. Secondly, we present some summation results for different kind of Neumann, Kapteyn and Schlömilch series built by Iν (x) and Lν (x) which are connected by a Sonin-Gubler formula, and by the associated modified Struve differential equation. Finally, solving a Fredholm type convolutional integral equation of the first kind, Bromwich-Wagner line integral expressions are derived for the Bessel function of the first kind Jν and for an associated generalized Schlömilch series.

Bounds for modified Struve functions of the first kind and their ratios

“…The modified Struve functions are related to the modified Bessel functions. Likewise, they arise in manyfold applications, including leakage inductance in transformer windings [16], perturbation approximations of lee waves in a stratified flow [29], scattering of plane waves by circular cylinders [44] and lift and downwash distributions of oscillating wings in subsonic and supersonic flow [45,46]; see [7] for a list of further application areas.…”

Inequalities for Integrals of the Modified Struve Function of the First Kind II

“…The function L ν (x) is itself a widely used special function; see a standard reference, such as [17], for its basic properties. It has numerous applications in the applied sciences, including leakage inductance in transformer windings [14], perturbation approximations of lee waves in a stratified flow [16], scattering of plane waves by soft obstacles [20]; see [1] for a list of further application areas. It is therefore a natural problem to ask for simple inequalities, involving the modified Struve function of the first kind, for the integrals…”

“…x 2 4 (see also [1] for other representations in terms of the generalized hypergeometric function). A straightforward calculation then yields…”

Applications of third order differential subordination and superordination involving generalized struve function

“…Struve functions are applied to various areas of applied mathematics and physics. In [10], and the reference therein various applications are illustrated in different context and also for its application in geometric function theory we refer [2], [8], [9], [11], [12], etc. Let S p denote the Struve function of order p is of the form…”