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The [[Integral transform|integral 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>
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$$ \tag{* }
 +
F( x)  = \int\limits _ { 0 } ^  \infty \frac{f(t)}{x+t} dt.
 +
$$
  
 
The Stieltjes transform arises in the iteration of the [[Laplace transform|Laplace transform]] and is also a particular case of a convolution transform.
 
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
+
One of the inversion formulas is as follows: If the function $  f( t) \sqrt t $
 +
is continuous and bounded on $  ( 0, \infty ) $,
 +
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>
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$$
 +
\lim\limits _ {n \rightarrow \infty } 
 +
\frac{(- 1)  ^ {n} }{2 \pi }
 +
\left (
 +
\frac{e}{n}
 +
\right )  ^ {2n} [ x
 +
^ {2n} F ^ { ( n) } ( x)]  ^ {(n)}  = f( x)
 +
$$
  
for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087810/s0878105.png" />.
+
for $  x \in ( 0, \infty ) $.
  
 
The generalized Stieltjes transform is
 
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>
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$$
 +
F( x)  = \int\limits _ { 0 } ^  \infty 
 +
\frac{f(t)}{( x+ t)  ^  \rho  } dt
 +
,
 +
$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087810/s0878107.png" /> is a complex number.
+
where $  \rho $
 +
is a complex number.
  
 
The integrated Stieltjes transform is
 
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>
+
$$
 +
F( x)  = \int\limits _ { 0 } ^  \infty  K( x, t) f( t)  dt,
 +
$$
  
 
where
 
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>
+
$$
 +
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).
 
Stieltjes transforms are also introduced for generalized functions. The transform (*) was studied by Th.J. Stieltjes (1894–1895).

Latest revision as of 14:00, 28 June 2020


The integral 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=49447
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