# Stieltjes transform

From Encyclopedia of Mathematics

(*) |

The Stieltjes transform arises in the iteration of the Laplace transform and is also a particular case of a convolution transform.

One of the inversion formulas is as follows: If the function is continuous and bounded on , then

for .

The generalized Stieltjes transform is

where is a complex number.

The integrated Stieltjes transform is

where

Stieltjes transforms are also introduced for generalized functions. The transform (*) was studied by Th.J. Stieltjes (1894–1895).

#### References

[1] | D.V. Widder, "The Laplace transform" , Princeton Univ. Press (1972) |

[2] | R.P. Boas, D.V. Widder, "The iterated Stieltjes transform" Trans. Amer. Math. Soc. , 45 (1939) pp. 1–72 |

[3] | E.C. Titchmarsh, "Introduction to the theory of Fourier integrals" , Oxford Univ. Press (1948) |

[4] | Y.A. Brychkov, A.P. Prudnikov, "Integral transforms of generalized functions" , Gordon & Breach (1989) (Translated from Russian) |

**How to Cite This Entry:**

Stieltjes transform. Yu.A. BrychkovA.P. Prudnikov (originator),

*Encyclopedia of Mathematics.*URL: http://www.encyclopediaofmath.org/index.php?title=Stieltjes_transform&oldid=17974

This text originally appeared in Encyclopedia of Mathematics - ISBN 1402006098