A simple derivation of the Fourier transform of the Heaviside function

TL;DR

This paper provides a straightforward, rigorous derivation of the Fourier transform of the Heaviside function using basic calculus concepts suitable for undergraduates.

Contribution

It introduces a simple derivation method within the tempered distributions framework, accessible to students with fundamental calculus knowledge.

Findings

Derivation relies on limits, generalized integrals, and integration by parts.

Framework suitable for undergraduate engineering and mathematics students.

Provides a rigorous proof accessible to beginners.

Abstract

We give a rigorous derivation of the Fourier transform of the Heaviside function within a framework for tempered distributions that is suitable for undergraduate engineering and mathematics students. The proofs rely on fundamental concepts typically taught in a freshman-level calculus course, including limits, generalized integrals, integration by parts and the Taylor Remainder Theorem.