View source for Stieltjes transform

The [[Integral transform|integral transform]]

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087810/s0878101.png" /></td> <td valign="top" style="width:5%;text-align:right;">(*)</td></tr></table>

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

One of the inversion formulas is as follows: If the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087810/s0878102.png" /> is continuous and bounded on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087810/s0878103.png" />, then

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087810/s0878104.png" /></td> </tr></table>

for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087810/s0878105.png" />.

The generalized Stieltjes transform is

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087810/s0878106.png" /></td> </tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087810/s0878107.png" /> is a complex number.

The integrated Stieltjes transform is

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087810/s0878108.png" /></td> </tr></table>

where

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087810/s0878109.png" /></td> </tr></table>

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

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  D.V. Widder,   "The Laplace transform" , Princeton Univ. Press  (1972)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  R.P. Boas,   D.V. Widder,   "The iterated Stieltjes transform"  ''Trans. Amer. Math. Soc.'' , '''45'''  (1939)  pp. 1–72</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  E.C. Titchmarsh,   "Introduction to the theory of Fourier integrals" , Oxford Univ. Press  (1948)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  Y.A. Brychkov,   A.P. Prudnikov,   "Integral transforms of generalized functions" , Gordon &amp; Breach  (1989)  (Translated from Russian)</TD></TR></table>