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Stieltjes Transform

The Stieltjes transform of a function f on the positive real line is the integral transform

S_f(z)=int_0^infty(f(t))/(z+t)dt,
(1)

wherever the improper integral exists, usually for z off the nonpositive real axis. A generalized Stieltjes transform of order p is

(K_pf)(z)=Gamma(p)int_0^infty(z+t)^(-p)f(t)dt.
(2)

Note the lower limit of 0, not -infty as implied in Samko et al. (1993, p. 23, eqn. 1.101).

If f is continuous at x>0, the standard transform can be inverted from its boundary value as

f(x)=-1/pilim_(epsilon->0^+)I[S_f(-x+iepsilon)].
(3)

A frequently used normalized Stieltjes kernel is

Phi(s)=int_0^infty(g(t))/(1+st)dt.
(4)

Writing M[h](q)=int_0^inftyx^(q-1)h(x)dx for the Mellin transform, interchange of the improper integrals gives

M[Phi](q)=picsc(piq)M[g](1-q),
(5)

for 0<R[q]<1 when the transforms converge (Ribeiro de Almeida et al. 2026).

See also

Laplace Transform, Mellin Transform

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References

Ribeiro de Almeida, R. R.; Lenzi, E. K.; and Evangelista, L. R. "Impedance Response of Electrolytic Cells with Non-Debye Relaxation Time Distributions at Adsorbing Electrodes." Electrochim. Acta 577, 149724, 2026. https://doi.org/10.1016/j.electacta.2026.149724.Samko, S. G.; Kilbas, A. A.; and Marichev, O. I. Fractional Integrals and Derivatives. Yverdon, Switzerland: Gordon and Breach, p. 23, 1993.Widder, D. V. The Laplace Transform. Princeton, NJ: Princeton University Press, 1941.

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Stieltjes Transform

Cite this as:

Weisstein, Eric W. "Stieltjes Transform." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/StieltjesTransform.html

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