# Stieltjes transform

$$\tag{* } F( x) = \int\limits _ { 0 } ^ \infty \frac{f(t)}{x+t} dt.$$

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 $f( t) \sqrt t$ is continuous and bounded on $( 0, \infty )$, then

$$\lim\limits _ {n \rightarrow \infty } \frac{(- 1) ^ {n} }{2 \pi } \left ( \frac{e}{n} \right ) ^ {2n} [ x ^ {2n} F ^ { ( n) } ( x)] ^ {(n)} = f( x)$$

for $x \in ( 0, \infty )$.

The generalized Stieltjes transform is

$$F( x) = \int\limits _ { 0 } ^ \infty \frac{f(t)}{( x+ t) ^ \rho } dt ,$$

where $\rho$ is a complex number.

The integrated Stieltjes transform is

$$F( x) = \int\limits _ { 0 } ^ \infty K( x, t) f( t) dt,$$

where

$$K( x, t) = \left \{ \begin{array}{ll} \frac{ \mathop{\rm ln} x / t }{x-t} , & t \neq x, \\ \frac{1}{x} , & t = x. \\ \end{array} \right .$$

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. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Stieltjes_transform&oldid=49826
This article was adapted from an original article by Yu.A. BrychkovA.P. Prudnikov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article