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On the relation between the Laplace and the Fourier transforms.

McGill University
Publication Date
  • Mathematics.
  • Mathematics


The subject of integral transforms is generally introduced as an extension of the Fourier series representation of periodic functions. The fundamental expression, the Fourier integral formula, is derived from these series by a purely formal process, and, from it, the Fourier transform and the inversion formula are defined. The validity of these expressions is then ascertained by the proof of the fundamental theorem of convergence for functions satisfying given conditions. The Laplace transform and its corresponding inversion formula are introduced, in such discussions, in an equally formal manner by the simple expedient of a definition.

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