Laplace-stieltjes Transform

Definition

$\backslash \{\backslash mathcal\{L\}^*g\backslash \}(s)\; =\; \backslash int\_\{-\backslash infty\}^\{\backslash infty\}\; \backslash mathrm\{e\}^\{-sx\}\backslash ,dg(x),\; \backslash quad\; s\; \backslash in\; \backslash mathbb\{C\},$

Properties

$\backslash \{\backslash mathcal\{L\}^*(g\; *\; h)\backslash \}(s)\; =\; \backslash \{\backslash mathcal\{L\}^*g\backslash \}(s)\backslash \{\backslash mathcal\{L\}^*h\backslash \}(s),$

Applications

$\backslash \{\backslash mathcal\{L\}^*F\backslash \}(s)\; =\; \backslash mathrm\{E\}\backslash left\backslash mathrm\{e\}^\{-sX\}\backslash right.$

Related Topics

• If g has derivative g' then the Laplace-Stieltjes transform of g is the Laplace transform of g' .

$\backslash \{\backslash mathcal\{L\}^*g\backslash \}(s)\; =\; \backslash \{\backslash mathcal\{L\}g\text{'}\backslash \}(s),$

• We can obtain the Fourier-Stieltjes transform of g (and, by the above note, the Fourier transform of g' ) by

$\backslash \{\backslash mathcal\{F\}^*g\backslash \}(s)\; =\; \backslash \{\backslash mathcal\{L\}^*g\backslash \}(\backslash mathrm\{i\}s),\; \backslash quad\; s\; \backslash in\; \backslash mathbb\{R\}.$

References

• Apostol, T.M. (1957). Mathematical Analysis. Addison-Wesley, Reading, MA. (For 1974 2nd ed, ISBN 0201002884).
• Apostol, T.M. (1997). Modular Functions and Dirichlet Series in Number Theory, 2nd ed. Springer-Verlag, New York. ISBN 0387971270.

• Grimmett, G.R. and Stirzaker, D.R. (2001). Probability and Random Processes, 3nd ed. Oxford University Press, Oxford. ISBN 0198572220.

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