# Laplace-Stieltjes transform

From Encyclopedia of Mathematics

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.

#### 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: http://www.encyclopediaofmath.org/index.php?title=Laplace-Stieltjes_transform&oldid=22709

This article was adapted from an original article by M. Hazewinkel (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article