Stieltjes transforms

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1: 2.6 Distributional Methods

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§2.6(ii) Stieltjes Transform

… ►The Stieltjes transform of $f(t)$ is defined by ►

2.6.8 $$𝒮f\left(z\right)={\int }_0^{\mathrm{\infty }}\frac{f(t)}{t+z}dt.$$

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Symbols:

$𝒮\left(f\right)\left(s\right)$: Stieltjes transform, $dx$: differential, $\int$: integral and $f(t)$: locally integrable function

Referenced by:

§2.6(ii)

Permalink:

http://dlmf.nist.gov/2.6.E8

Encodings:

pMML, png, TeX

Notational Change (effective with 1.0.15):

The notation for the Stieltjestransform was changed to $𝒮f\left(z\right)$ from $𝒮(f;z)$.

See also:

Annotations for §2.6(ii), §2.6 and Ch.2

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2.6.25 $$\underset{\epsilon \to 0}{lim}⟨f,{\varphi }_{\epsilon }⟩=𝒮f\left(z\right),$$

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Symbols:

$𝒮\left(f\right)\left(s\right)$: Stieltjes transform, $⟨\mathrm{\Lambda },\varphi ⟩$: inner-product of distribution, $\epsilon$: small positive number and $f(t)$: locally integrable function

Permalink:

http://dlmf.nist.gov/2.6.E25

Encodings:

pMML, png, TeX

Notational Change (effective with 1.0.15):

The notation for the Stieltjestransform was changed to $𝒮f\left(z\right)$ from $𝒮(f;z)$.

See also:

Annotations for §2.6(ii), §2.6 and Ch.2

… ►Corresponding results for the generalized Stieltjes transform …

2: 1.14 Integral Transforms

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§1.14(vi) Stieltjes Transform

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1.14.47 $$𝒮\left(f\right)\left(s\right)=𝒮f\left(s\right)={\int }_0^{\mathrm{\infty }}\frac{f(t)}{s+t}dt.$$

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Defines:

$𝒮\left(f\right)\left(s\right)$: Stieltjes transform

Symbols:

$dx$: differential and $\int$: integral

Referenced by:

§1.14(vi), §1.14(vi)

Permalink:

http://dlmf.nist.gov/1.14.E47

Encodings:

pMML, png, TeX

Notational Change (effective with 1.0.15):

The notation for the Stieltjestransform was changed to $𝒮\left(f\right)\left(s\right)$ or $𝒮f\left(s\right)$ from $𝒮(f;s)$.

See also:

Annotations for §1.14(vi), §1.14 and Ch.1

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Inversion

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Laplace Transform

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3: 9.17 Methods of Computation

… ►Among the integral representations of the Airy functions the Stieltjes transform (9.10.18) furnishes a way of computing $\mathrm{Ai}\left(z\right)$ in the complex plane, once values of this function can be generated on the positive real axis. …

4: 9.10 Integrals

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§9.10(vii) Stieltjes Transforms

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5: Bibliography N

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On the large argument asymptotics of the Lommel function via Stieltjes transforms. Asymptot. Anal. 91 (3-4), pp. 265–281.

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External Links:

ISSN 0921-7134, Document, MathReview Entry, ZentralBlatt

Referenced by:

§11.6(iii), §11.6(iii), §11.9(iii), §11.9(iii).

Encodings:

BibTeX

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6: Bibliography J

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Asymptotic behavior of the continued fraction coefficients of a class of Stieltjes transforms including the Binet function. In Orthogonal functions, moment theory, and continued fractions (Campinas, 1996), Lecture Notes in Pure and Appl. Math., Vol. 199, pp. 257–274.

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External Links:

MathReview (Olav Njåstad), ZentralBlatt

Referenced by:

§5.10, §5.10.

Encodings:

BibTeX

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7: Errata

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There have been extensive changes in the notation used for the integral transforms defined in §1.14. These changes are applied throughout the DLMF. The following table summarizes the changes.

Transform	New	Abbreviated	Old
	Notation	Notation	Notation
Fourier	$ℱ\left(f\right)\left(x\right)$	$ℱf\left(x\right)$
Fourier Cosine	${ℱ}_c\left(f\right)\left(x\right)$	${ℱ}_{c}f\left(x\right)$
Fourier Sine	${ℱ}_s\left(f\right)\left(x\right)$	${ℱ}_{s}f\left(x\right)$
Laplace	$ℒ\left(f\right)\left(s\right)$	$ℒf\left(s\right)$	$ℒ(f(t);s)$
Mellin	$ℳ\left(f\right)\left(s\right)$	$ℳf\left(s\right)$	$ℳ(f;s)$
Hilbert	$\mathscr{H}\left(f\right)\left(s\right)$	$\mathscr{H}f\left(s\right)$	$\mathscr{H}(f;s)$
Stieltjes	$𝒮\left(f\right)\left(s\right)$	$𝒮f\left(s\right)$	$𝒮(f;s)$

Previously, for the Fourier, Fourier cosine and Fourier sine transforms, either temporary local notations were used or the Fourier integrals were written out explicitly.

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8: Bibliography W

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Gevrey asymptotics and Stieltjes transforms of algebraically decaying functions. Proc. Roy. Soc. London Ser. A 458, pp. 625–644.

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External Links:

ISSN 1364-5021, Document, MathReview, ZentralBlatt

Referenced by:

§2.11(v).

Encodings:

BibTeX

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9: Bibliography K

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Fractional integral and generalized Stieltjes transforms for hypergeometric functions as transmutation operators. SIGMA Symmetry Integrability Geom. Methods Appl. 11, pp. Paper 074, 22.

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External Links:

ISSN 1815-0659, Document, Link, MathReview Entry, ZentralBlatt

Referenced by:

§15.14, §15.14.

Encodings:

BibTeX

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10: Bibliography B

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Stieltjes transforms and the Stokes phenomenon. Proc. Roy. Soc. London Ser. A 429, pp. 227–246.

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External Links:

ISSN 0962-8444, FullText, MathReview, ZentralBlatt

Referenced by:

§2.11(v).

Encodings:

BibTeX

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