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3 Journal of Applied Analysis

3 Patkowski Alexander E.

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A general size-biased distribution

Patkowski Alexander E.

Journal of Applied Analysis

2023

Vol. 29, nr 2

347--351

We generalize a size-biased distribution related to the Riemann xi-function using the work of Ferrar. Some analysis and properties of this more general distribution are offered as well.

On a solution to a functional equation

Patkowski Alexander E.

Journal of Applied Analysis

2022

Vol. 28, nr 1

91--93

We offer a solution to a functional equation using properties of the Mellin transform. A new criteria for the Riemann Hypothesis is offered as an application of our main result, through a functional relationship with the Riemann xi function.

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.