Fourier Transforms in Two or More Variables

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

Abstract

The theory of Fourier transforms of a single variable may be extended to functions of several variables. Thus, if f(x,y) is a function of two variables, the function F(ξ, η) defined by
(1)
is the two-dimensional Fourier transform of f(x,y), and, provided that the inversion formula (7.6) may be applied twice, we have
(2)
.

Keywords

Fourier Transform Radiation Condition Simple Algebra Fresnel Diffraction Fraunhofer Diffraction 
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.

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Footnotes

  1. 4.
    This result is given in I. N. Sneddon, J. Eng. Math. (1974), 8, 177, together with a discussion of the connection with the half-space Dirichlet problem for Laplace’s equation.Google Scholar
  2. 6.
    See DITKIN & PRUDNIKOV (1970) for more information on double Laplace transforms.Google Scholar
  3. 8.
    See J. C. Jaeger, Bull. Am. Math. Soc. (1940), 46, 687.MathSciNetCrossRefGoogle Scholar
  4. 9.
    The application of the double Laplace transform to a more general second-order partial differential equation in the quadrant x 2265; 0, y 2265; 0 is discussed in K. Evans and E. A. Jackson, J. Math. Phys. (1971), 12, 2012.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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