# Laplace-Stieltjes transform

Let $G ( t )$ be a function of bounded variation on $0 \leq t \leq R$, for all positive $R$. The integral

$$f ( s ) = \int\limits _ { 0 } ^ \infty {e ^ {- st } } {dG ( t ) } = {\lim\limits } _ {R \rightarrow \infty } \int\limits _ { 0 } ^ { R } {e ^ {- st } } {dG ( t ) }$$

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

If the integral converges for some complex number $s _ {0}$, then it converges for all $s$ with ${ \mathop{\rm Re} } ( s ) > { \mathop{\rm Re} } ( s _ {0} )$, and the function $f ( s )$ is then the Laplace–Stieltjes transform of $G$. If $G$ is of the form $G ( t ) = \int _ {0} ^ {t} {g ( t ) } {dt }$ for a function $g$ on $[ 0,t ]$ that is Lebesgue integrable for all $t$( see Lebesgue integral), then the Laplace–Stieltjes transform becomes the Laplace transform $f ( s ) = \int _ {0} ^ \infty {e ^ {- st } g ( t ) } {dt }$ of $g$.

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

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://encyclopediaofmath.org/index.php?title=Laplace-Stieltjes_transform&oldid=47577
This article was adapted from an original article by M. Hazewinkel (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article