pISSN 0374-4914 eISSN 2289-0041

## Research Paper

New Phys.: Sae Mulli 2020; 70: 759-765

Published online September 29, 2020 https://doi.org/10.3938/NPSM.70.759

## General Definitions of Integral Transforms for Mathematical Physics

Dongseung KANG1, Hoewoon KIM2, Bongwoo LEE*3

1Department of Mathematics Education, Dankook University, Gyeonggi 16890, Korea

2Department of Mathematics, Oregon State University, Corvallis, Oregon 97331, USA
3Department of Science Education, Dankook University, Gyeonggi 16890, Korea

Correspondence to:peak@dankook.ac.kr

Received: July 10, 2020; Revised: August 12, 2020; Accepted: August 18, 2020

### Abstract

The Laplace and the Fourier transforms are famous integral transform methods in mathematical physics for solving differential equations. However, most undergraduate textbooks on differential equations only contain a few chapters covering Laplace and Fourier transforms, begin with their definitions as improper integrals on a half line, $(0,\infty)$, or on an entire line, $(-\infty,\infty)$, respectively, proceed with their properties, and then apply them to solve differential equations. Many students just follow these procedures while wondering how those integral transform methods arise naturally for physically reasonable situations. This present article presents new perspectives on derive the general definitions of the Laplace and the Fourier transforms and presents examples in physics to help students discover the Laplace and the Fourier transforms on various domains, in contrast to only a half line and a whole line in most textbooks.

Keywords: Laplace transform, Fourier transform, Integral transforms, Mathematical physics, Differential equations, Undergraduate mathematics