In these cases, the image of the Laplace transform lives in a space of analytic functions in the region of convergence. The Laplace transform is also defined and injective for suitable spaces of tempered distributions. Typical function spaces in which this is true include the spaces of bounded continuous functions, the space L ∞(0, ∞), or more generally tempered distributions on (0, ∞). In fact, besides integrable functions, the Laplace transform is a one-to-one mapping from one function space into another in many other function spaces as well, although there is usually no easy characterization of the range. This means that, on the range of the transform, there is an inverse transform. Two integrable functions have the same Laplace transform only if they differ on a set of Lebesgue measure zero.

The advantages of the Laplace transform had been emphasized by Gustav Doetsch, to whom the name Laplace transform is apparently due.įrom 1744, Leonhard Euler investigated integrals of the form The current widespread use of the transform (mainly in engineering) came about during and soon after World War II, replacing the earlier Heaviside operational calculus. The theory was further developed in the 19th and early 20th centuries by Mathias Lerch, Oliver Heaviside, and Thomas Bromwich.

Laplace's use of generating functions was similar to what is now known as the z-transform, and he gave little attention to the continuous variable case which was discussed by Niels Henrik Abel. Laplace wrote extensively about the use of generating functions in Essai philosophique sur les probabilités (1814), and the integral form of the Laplace transform evolved naturally as a result. The Laplace transform is named after mathematician and astronomer Pierre-Simon, marquis de Laplace, who used a similar transform in his work on probability theory.

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