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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, ∞), or on an entire line, (−∞, ∞), 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.
목차
Ⅰ. Introduction
Ⅱ. General Definitions of Laplace and Fourier Transforms
Ⅲ. Laplace Transforms and Fourier Transforms on Various Domains
Ⅳ. Conclusion
REFERENCES
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