Access to eBooks is limited to institutions that have purchased
or currently subscribe to the SPIE eBooks program. eBooks are not
available via an individual subscription. SPIE books (print and
digital) may be purchased individually on
Contact your librarian to recommend SPIE eBooks for your organization.
Chapter 2: Fourier Integrals and Fourier Transforms
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.
Online access to SPIE eBooks is limited to subscribing institutions.