Laplace-Stieltjes transform

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Let be a function of bounded variation on , for all positive . The integral

is known as a (formal) Laplace–Stieltjes integral.

If the integral converges for some complex number , then it converges for all with , and the function is then the Laplace–Stieltjes transform of . If is of the form for a function on that is Lebesgue integrable for all (see Lebesgue integral), then the Laplace–Stieltjes transform becomes the Laplace transform of .

There is also a corresponding two-sided Laplace–Stieltjes transform (or bilateral Laplace–Stieltjes transform) for suitable functions .

See Laplace transform for additional references.


[a1] D.V. Widder, "An introduction to transform theory" , Acad. Press (1971)
How to Cite This Entry:
Laplace-Stieltjes transform. Encyclopedia of Mathematics. URL:
This article was adapted from an original article by M. Hazewinkel (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article