Definitions and Elementary Properties

  • B. Davies
Part of the Applied Mathematical Sciences book series (AMS, volume 25)

Abstract

Let f(t) be an arbitrary function; then the (exponential) Fourier transform of f(t) is the function defined by the integral
$$ F\left( \omega \right) = \int_{ - \infty }^\infty {{e^{i\omega t}}f\left( t \right)dt} $$
(1)
for those values of ω for which the integral exists.

Keywords

Fourier Transform Inverse Fourier Transform Elementary Property Poisson Summation Formula Fourier Cosine 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Footnotes

  1. 1.
    This method leaves the value of f(0) undetermined, a matter of no practical consequence since inverse transforms are unique only to within a null function.Google Scholar
  2. 2.
    Alternatively, we could appeal to Bessel’s integral (20.50) immediately to obtain the result (19).Google Scholar
  3. 3.
    We will see in Section 9 that (25) and (26) are valid for generalized functions with no additional assumptions.Google Scholar
  4. 4.
    In quantum mechanics, this is the uncertainty principle.Google Scholar

Copyright information

© Springer Science+Business Media New York 1978

Authors and Affiliations

  • B. Davies
    • 1
  1. 1.The Australian National UniversityCanberraAustralia

Personalised recommendations