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) |
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