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)