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
is the two-dimensional Fourier transform of f(x,y), and, provided that the inversion formula (7.6) may be applied twice, we have
.
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Footnotes
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.
See DITKIN & PRUDNIKOV (1970) for more information on double Laplace transforms.
See J. C. Jaeger, Bull. Am. Math. Soc. (1940), 46, 687.
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.
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© 1978 Springer Science+Business Media New York
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Davies, B. (1978). Fourier Transforms in Two or More Variables. In: Integral Transforms and Their Applications. Applied Mathematical Sciences, vol 25. Springer, New York, NY. https://doi.org/10.1007/978-1-4757-5512-1_11
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DOI: https://doi.org/10.1007/978-1-4757-5512-1_11
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