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On integrals associated with the free particle wave packet

Patkowski Alexander E.

Journal of Applied Analysis

2020

Vol. 26, nr 2

257--262

We discuss some properties of integrals associated with the free particle wave packet, ψ(x, t), which are solutions to the time-dependent Schrödinger equation for a free particle. Some noteworthy discussion is made in relation to integrals which have appeared in the literature. We also obtain formulas for half-integer arguments of the Riemann zeta function.

On some non-Gaussian wave packets

Patkowski A. E.

Journal of Applied Analysis

2018

Vol. 24, nr 1

109--114

We expand on a previous study by offering a generalized wave function associated with the parabolic cylinder function and a connection with a two-particle position-space wave function. We also provide an explicit formula for a wave function associated with a recent work by the present author and M. Wolf.

The fracture mechanics formulas for split-tension strips

Ince R.

Journal of Theoretical and Applied Mechanics

2017

Vol. 55 nr 2

607--619

The notched beams have been commonly used in concrete fracture. In this study, the splitting-strip specimens, which have some advantages – compactness and lightness – compared to beams, are analyzed for effective crack models. Using the Fourier integrals and Fourier series, a formula for the maximum tensile strength of concrete is first derived for an un-notched splitting-strip in the plane of loading. Subsequently, the linear elastic fracture mechanics formulas of the splitting-strip specimens, namely the stress intensity factor KI , the crack mouth opening displacement CMOD, and the crack opening displacement profile COD, are determined for different load-distributed widths via the weight function.

Some Remarks on Glaisher-Ramanujan Type Integrals

Patkowski A. E., Wolf M.

Computational Methods in Science and Technology

2016

Vol. 22, No. 2

103--108

Some integrals of the Glaisher-Ramanujan type are established in a more general form than in previous studies. As an application we prove some Ramanujan-type series identities, as well as a new formula for the Dirichlet beta function at the value s = 3. We also present a Mathematica program calculating values of beta function at odd positive arguments.

A radial version of the Kontorovich-Lebedev transform in the unit ball

Yakubovich S. B., Vieira N.

Opuscula Mathematica

2011

Vol. 31, no. 1

137-146

In this paper we introduce a radial version of the Kontorovich-Lebedev transform in the unit ball. Mapping properties and an inversion formula are proved in Lp.