Stieltjes transform

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)