Fourier Transforms in Two or More Variables

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


The theory of Fourier transforms of a single variable may be extended to functions of several variables.


Fourier Transform Radiation Condition Simple Algebra Fresnel Diffraction Fraunhofer Diffraction 
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  1. 1.
    These results apply either to functions having the necessary behavior at infinity to allow integration by parts, or to generalized functions with no restrictions.Google Scholar
  2. 2.
    The theory of generalized functions may be extended quite simply to several variables, but we do not need to concern ourselves with the details here.Google Scholar
  3. 3.
    See Section 9.5.Google Scholar
  4. 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.MathSciNetzbMATHCrossRefGoogle Scholar
  5. 5.
    This is an example of the collisionless linear transport equation. See Section 19.6 for an example of the use of this Green’s function in the solution of the linear transport equation with collisions.Google Scholar
  6. 6.
    See Ditkin & Prudnikov (1970) for more information on double Laplace transforms.Google Scholar
  7. 7.
    We use the notation ℒ[f (x,y); y → p] so as to indicate which variable is transformed. Thus ℒ[f (x,y); y → p] is a function of x and p.Google Scholar
  8. 8.
    See J. C. Jaeger, Bull. Am. Math. Soc. (1940), 46, 687.MathSciNetCrossRefGoogle Scholar
  9. 9.
    The application of the double Laplace transform to a more general second-order partial differential equation in the quadrant x ≥ 0, y ≥ 0 is discussed in K. Evans and E. A. Jackson, J. Math. Phys. (1971), 12, 2012.MathSciNetzbMATHCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 1985

Authors and Affiliations

  • B. Davies
    • 1
  1. 1.Department of MathematicsThe Australian National UniversityCanberraAustralia

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