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1: 2.6 Distributional Methods
§2.6(ii) Stieltjes Transform
The Stieltjes transform of f ( t ) is defined by
2.6.8 𝒮 f ( z ) = 0 f ( t ) t + z d t .
2.6.25 lim ε 0 f , ϕ ε = 𝒮 f ( z ) ,
Corresponding results for the generalized Stieltjes transform
2: 1.14 Integral Transforms
§1.14(vi) Stieltjes Transform
1.14.47 𝒮 ( f ) ( s ) = 𝒮 f ( s ) = 0 f ( t ) s + t d t .
Inversion
Laplace Transform
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 Ai ( z ) in the complex plane, once values of this function can be generated on the positive real axis. …
4: 9.10 Integrals
§9.10(vii) Stieltjes Transforms
5: Bibliography N
  • G. Nemes (2015b) On the large argument asymptotics of the Lommel function via Stieltjes transforms. Asymptot. Anal. 91 (3-4), pp. 265–281.
  • 6: Bibliography J
  • W. B. Jones and W. Van Assche (1998) 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.
  • 7: Errata
  • Section 1.14

    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 ) ( x ) f ( x )
    Fourier Cosine c ( f ) ( x ) c f ( x )
    Fourier Sine s ( f ) ( x ) s f ( x )
    Laplace ( f ) ( s ) f ( s ) ( f ( t ) ; s )
    Mellin ( f ) ( s ) f ( s ) ( f ; s )
    Hilbert ( f ) ( s ) f ( s ) ( f ; s )
    Stieltjes 𝒮 ( f ) ( s ) 𝒮 f ( s ) 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.

  • 8: Bibliography W
  • R. Wong and Y. Zhao (2002b) Gevrey asymptotics and Stieltjes transforms of algebraically decaying functions. Proc. Roy. Soc. London Ser. A 458, pp. 625–644.
  • 9: Bibliography K
  • T. H. Koornwinder (2015) Fractional integral and generalized Stieltjes transforms for hypergeometric functions as transmutation operators. SIGMA Symmetry Integrability Geom. Methods Appl. 11, pp. Paper 074, 22.
  • 10: Bibliography B
  • W. G. C. Boyd (1990b) Stieltjes transforms and the Stokes phenomenon. Proc. Roy. Soc. London Ser. A 429, pp. 227–246.