Definitions and Elementary Properties

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


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


Fourier Transform Inverse Fourier Transform Elementary Property Poisson Summation Formula Fourier Cosine 
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  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

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