Fourier Integrals and Fourier Transforms

doi:10.1117/3.339204.ch2

Book: Integral Transforms for Engineers

Author(s): Larry C. Andrews, Bhimsen K. Shivamoggi

Published: 1999

https://doi.org/10.1117/3.339204.ch2

Author Affiliations +

Abstract

The concept of an infinite series dates back as far as the ancient Greeks such as Archimedes (287-212 b.c., who summed a geometric series in order to compute the area under a parabolic arc. In the eighteenth century, power series expansions for functions like e x , sin x, and arctan x were first published by the Scottish mathematician C. Maclaurin (1698-1746), and British mathematician B. Taylor (1685-1731) generalized this work by providing power series expansions about some point other than x=0 . By the middle of the eighteenth century it became important to study the possibility of representing a given function by infinite series other than power series. D. Bernoulli (1700-1783) showed that the mathematical conditions imposed by physical considerations in solving the vibrating-string problem were formally satisfied by functions represented as infinite series involving sinusoidal functions.