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 of $x$ , $\int$: 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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2.6.28 $$𝒮f\left(z\right)=\ln z\sum _{s=0}^{n-1}{(-1)}^s\frac{a_s}{z^{s+1}}+\sum _{s=0}^{n-1}{(-1)}^s\frac{{\tilde{d}}_s}{z^{s+1}}+R_n(z),$$

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Symbols:

$𝒮\left(f\right)\left(s\right)$: Stieltjes transform, $\ln z$: principal branch of logarithm function, $R_n(z)$: remainder, $f(t)$: locally integrable function, $a_n$: coefficients and $n$: nonnegative integer

Permalink:

http://dlmf.nist.gov/2.6.E28

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 of $x$ 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: 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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8: Errata

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The following additions were made in Chapter 1:

• New subsections, 1.2(v) Matrices, Vectors, Scalar Products, and Norms and 1.2(vi) Square Matrices, with Equations (1.2.27)–(1.2.77).

• The title of this section was changed from “Determinants” to “Determinants, Linear Operators, and Spectral Expansions”. An extra paragraph just below (1.3.7). New subsection, 1.3(iv) Matrices as Linear Operators, with Equations (1.3.20), (1.3.21).

• In 1.4(v), three new paragraphs on Stieltjes integrals/measure, with Equations (1.4.23_1)–(1.4.23_3).

• In Subsection 1.8(i), the title of the paragraph “Bessel’s Inequality” was changed to “Parseval’s Formula”. We give the relation between the real and the complex coefficients, and include more general versions of Parseval’s Formula, Equations (1.8.6_1), (1.8.6_2). The title of Subsection 1.8(iv) was changed from “Transformations” to “Poisson’s Summation Formula”, and we added an extra remark just below (1.8.14).

• New subsection, 1.10(xi) Generating Functions, with Equations (1.10.26)–(1.10.29).

• New subsection, 1.13(viii) Eigenvalues and Eigenfunctions: Sturm-Liouville and Liouville forms, with Equations (1.13.26)–(1.13.31).

• Another form of Parseval’s formula, (1.14.7_5).

• We include several extra remarks and Equations (1.16.3_5), (1.16.9_5). New subsection, 1.16(ix) References for Section 1.16.

• Two extra paragraphs in Subsection 1.17(ii) Integral Representations, with Equations (1.17.12_1), (1.17.12_2); Subsection 1.17(iv) Mathematical Definitions is almost completely rewritten.

• An entire new section, 1.18 Linear Second Order Differential Operators and Eigenfunction Expansions, including new subsections, 1.18(i)–1.18(x), and several equations, (1.18.1)–(1.18.71).

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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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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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