### Journals Information

**
Mathematics and Statistics Vol. 9(5), pp. 724 - 735 DOI: 10.13189/ms.2021.090512 Reprint (PDF) (266Kb) **

## Mellin Transform of an Exponential Fourier Transform Expressed in Terms of the Lerch Function

**Robert Reynolds ^{*}, Allan Stauffer **

Department of Mathematics and Statistics, Faculty of Science, York University, Canada

**ABSTRACT**

The aim of this paper is to provide a table of definite integrals which includes both known and new integrals. This work is important because we provide a formal derivation for integrals in [7] not currently present in literature along with new integrals. By deriving new integrals we hope to expand the current list of integral formulae which could assist in research where applicable. The authors apply their contour integral method [9] to an integral in [8] to achieve this new integral formula in terms of the Lerch function. In this present work, the authors provide a formal derivation for an interesting Exponential Fourier transform and express it in terms of the Lerch function. The Exponential Fourier transform has many real world applications namely, in the field of Electrical engineering, in the work of electrical transients by [10] and in the field of Civil engineering, in the work of stress analysis of boundary load on soil by [11]. The definite integral we derived in this work is given by (1) where the variables . This formal derivation is then used to derive the correct version of a definite integral transform along with new formulae. Some of the results in this work are new.

**KEYWORDS**

Hilbert Transform, Mellin Transform, Exponential Fourier Transform, Hyperbolic Function, Catalan's Constant

**Cite This Paper in IEEE or APA Citation Styles**

(a). IEEE Format:

[1] Robert Reynolds , Allan Stauffer , "Mellin Transform of an Exponential Fourier Transform Expressed in Terms of the Lerch Function," Mathematics and Statistics, Vol. 9, No. 5, pp. 724 - 735, 2021. DOI: 10.13189/ms.2021.090512.

(b). APA Format:

Robert Reynolds , Allan Stauffer (2021). Mellin Transform of an Exponential Fourier Transform Expressed in Terms of the Lerch Function. Mathematics and Statistics, 9(5), 724 - 735. DOI: 10.13189/ms.2021.090512.