integral transforms in terms of

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31—40 of 48 matching pages

31: Bibliography B

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P. Baratella and L. Gatteschi (1988) The Bounds for the Error Term of an Asymptotic Approximation of Jacobi Polynomials. In Orthogonal Polynomials and Their Applications (Segovia, 1986), Lecture Notes in Math., Vol. 1329, pp. 203–221.

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External Links:: MathReview (F. W. J. Olver), ZentralBlatt
Referenced by:: §18.15(i), §18.15(i).
Encodings:: BibTeX

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S. Bielski (2013) Orthogonality relations for the associated Legendre functions of imaginary order. Integral Transforms Spec. Funct. 24 (4), pp. 331–337.

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External Links:: ISSN 1065-2469, Document, FullText, MathReview (Praveen Agarwal), ZentralBlatt
Referenced by:: §14.17(iii), §14.17(iii).
Encodings:: BibTeX

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B. L. J. Braaksma and B. Meulenbeld (1967) Integral transforms with generalized Legendre functions as kernels. Compositio Math. 18, pp. 235–287.

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External Links:: MathReview (R. K. Saxena), ZentralBlatt
Referenced by:: §14.20(vi), §14.29, §14.31(ii).
Encodings:: BibTeX

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R. Bulirsch (1969a) An extension of the Bartky-transformation to incomplete elliptic integrals of the third kind. Numer. Math. 13 (3), pp. 266–284.

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External Links:: ISSN 0029-599X, Document, MathReview (Y. L. Luke), ZentralBlatt
Referenced by:: §19.1, §19.2(iii), §19.36(ii), §19.36(ii).
Encodings:: BibTeX

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J. G. Byatt-Smith (2000) The Borel transform and its use in the summation of asymptotic expansions. Stud. Appl. Math. 105 (2), pp. 83–113.

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External Links:: ISSN 0022-2526, Document, MathReview (A. D. Wood), ZentralBlatt
Referenced by:: §2.11(v).
Encodings:: BibTeX

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32: 9.10 Integrals

… ►For higher terms in (9.10.4)–(9.10.7) see Vallée and Soares (2010, §3.1.3). … ►

§9.10(v) Laplace Transforms

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§9.10(vi) Mellin Transform

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§9.10(vii) Stieltjes Transforms

… ►For further integrals, including the Airy transform, see §9.11(iv), Widder (1979), Prudnikov et al. (1990, §1.8.1), Prudnikov et al. (1992a, pp. 405–413), Prudnikov et al. (1992b, §4.3.25), Vallée and Soares (2010, Chapters 3, 4).

33: Bibliography

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M. J. Ablowitz and H. Segur (1981) Solitons and the Inverse Scattering Transform. SIAM Studies in Applied Mathematics, Vol. 4, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA.

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External Links:: ISBN 0-89871-174-6, MathReview (Benno Fuchssteiner), ZentralBlatt
Referenced by:: §21.9, §21.9, §32.5, §9.16, Profile Mark J. Ablowitz.
Encodings:: BibTeX

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M. Abramowitz (1954) Regular and irregular Coulomb wave functions expressed in terms of Bessel-Clifford functions. J. Math. Physics 33, pp. 111–116.

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External Links:: MathReview (E. Weber), ZentralBlatt
Referenced by:: §33.9(ii).
Encodings:: BibTeX

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N. I. Akhiezer (1988) Lectures on Integral Transforms. Translations of Mathematical Monographs, Vol. 70, American Mathematical Society, Providence, RI.

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Notes:: Translated from the Russian by H. H. McFaden
External Links:: ISBN 0-8218-4524-1, MathReview, ZentralBlatt
Referenced by:: §10.22(v).
Encodings:: BibTeX

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S. V. Aksenov, M. A. Savageau, U. D. Jentschura, J. Becher, G. Soff, and P. J. Mohr (2003) Application of the combined nonlinear-condensation transformation to problems in statistical analysis and theoretical physics. Comput. Phys. Comm. 150 (1), pp. 1–20.

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Notes:: Includes algorithms for Lerch’s transcendent. C and Mathematica codes by Aksenov and Jentschura are available.
External Links:: Code, Document
Referenced by:: 1st item.
Encodings:: BibTeX

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G. E. Andrews (2001) Bailey’s Transform, Lemma, Chains and Tree. In Special Functions 2000: Current Perspective and Future Directions (Tempe, AZ), J. Bustoz, M. E. H. Ismail, and S. K. Suslov (Eds.), NATO Sci. Ser. II Math. Phys. Chem., Vol. 30, pp. 1–22.

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External Links:: ISBN 0-7923-7119-4, MathReview, ZentralBlatt
Referenced by:: §17.12.
Encodings:: BibTeX

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34: 1.17 Integral and Series Representations of the Dirac Delta

… ►From the mathematical standpoint the left-hand side of (1.17.2) can be interpreted as a generalized integral in the sense that … ►Formal interchange of the order of integration in the Fourier integral formula ((1.14.1) and (1.14.4)): … ►Other similar integral representations of the Dirac delta that appear in the physics and applied mathematics literature include the following: … ►Integral representation (1.17.12_1), (1.17.12_2) is the equivalent of the transform pairs, (1.14.9)

\&

(1.14.11), (1.14.10)

\&

(1.14.12), respectively. … ►Equations (1.17.12_1) through (1.17.16) may re-interpreted as spectral representations of completeness relations, expressed in terms of Dirac delta distributions, as discussed in §1.18(v), and §1.18(vi) Further mathematical underpinnings are referenced in §1.17(iv). …

35: 22.18 Mathematical Applications

… ►With

k\in[0,1]

the mapping

z\to w=\operatorname{sn}\left(z,k\right)

gives a conformal map of the closed rectangle

[-K,K]\times[0,K^{\prime}]

onto the half-plane

\Im w\geq 0

, with

0,\pm K,\pm K+iK^{\prime},iK^{\prime}

mapping to

0,\pm 1,\pm k^{-2},\infty

respectively. The half-open rectangle

(-K,K)\times[-K^{\prime},K^{\prime}]

maps onto

\mathbb{C}

cut along the intervals

(-\infty,-1]

and

[1,\infty)

. … ►in which

a, b, c, d, e, f

are real constants, can be achieved in terms of single-valued functions. …Discussion of parametrization of the angles of spherical trigonometry in terms of Jacobian elliptic functions is given in Greenhill (1959, p. 131) and Lawden (1989, §4.4). … ►With the identification

x=\operatorname{sn}\left(z,k\right)

y=\ifrac{\mathrm{d}(\operatorname{sn}\left(z,k\right))}{\mathrm{d}z}

, the addition law (22.18.8) is transformed into the addition theorem (22.8.1); see Akhiezer (1990, pp. 42, 45, 73–74) and McKean and Moll (1999, §§2.14, 2.16). …

36: 19.33 Triaxial Ellipsoids

… ►Application of (19.16.23) transforms the last quantity in (19.30.5) into a two-dimensional analog of (19.33.1). … ►

§19.33(ii) Potential of a Charged Conducting Ellipsoid

… ►The same result holds for a homogeneous dielectric ellipsoid in an electric field. …Expressions in terms of Legendre’s integrals, numerical tables, and further references are given by Cronemeyer (1991). … ►In suitable units the self-energy of the distribution is given by …

37: Errata

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Expansion

§4.13 has been enlarged. The Lambert $W$ -function is multi-valued and we use the notation $W_{k}\left(x\right)$ , $k\in\mathbb{Z}$ , for the branches. The original two solutions are identified via $\operatorname{Wp}\left(x\right)=W_{0}\left(x\right)$ and $\operatorname{Wm}\left(x\right)=W_{\pm 1}\left(x\mp 0\mathrm{i}\right)$ .

Other changes are the introduction of the Wright $\omega$ -function and tree $T$ -function in (4.13.1_2) and (4.13.1_3), simplification formulas (4.13.3_1) and (4.13.3_2), explicit representation (4.13.4_1) for $\frac{{\mathrm{d}}^{n}W}{{\mathrm{d}z}^{n}}$ , additional Maclaurin series (4.13.5_1) and (4.13.5_2), an explicit expansion about the branch point at $z=-{\mathrm{e}}^{-1}$ in (4.13.9_1), extending the number of terms in asymptotic expansions (4.13.10) and (4.13.11), and including several integrals and integral representations for Lambert $W$ -functions in the end of the section.

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Subsection 18.15(i)

In the line just below (18.15.4), it was previously stated “is less than twice the first neglected term in absolute value.” It now states “is less than twice the first neglected term in absolute value, in which one has to take $\cos\theta_{n,m,\ell}=1$ .”

Reported by Gergő Nemes on 2019-02-08

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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	$\mathscr{F}\left(f\right)\left(x\right)$	$\mathscr{F}\mskip-3.0muf\mskip 3.0mu\left(x\right)$
Fourier Cosine	$\mathscr{F}_{\mkern-3.0muc}\left(f\right)\left(x\right)$	$\mathscr{F}_{\mkern-3.0muc}\mskip-1.0muf\mskip 3.0mu\left(x\right)$
Fourier Sine	$\mathscr{F}_{\mkern-2.0mus}\left(f\right)\left(x\right)$	$\mathscr{F}_{\mkern-2.0mus}\mskip-1.0muf\mskip 3.0mu\left(x\right)$
Laplace	$\mathscr{L}\left(f\right)\left(s\right)$	$\mathscr{L}\mskip-3.0muf\mskip 3.0mu\left(s\right)$	$\mathscr{L}(f(t);s)$
Mellin	$\mathscr{M}\left(f\right)\left(s\right)$	$\mathscr{M}\mskip-3.0muf\mskip 3.0mu\left(s\right)$	$\mathscr{M}(f;s)$
Hilbert	$\mathcal{H}\left(f\right)\left(s\right)$	$\mathcal{H}\mskip-3.0muf\mskip 3.0mu\left(s\right)$	$\mathcal{H}(f;s)$
Stieltjes	$\mathcal{S}\left(f\right)\left(s\right)$	$\mathcal{S}\mskip-3.0muf\mskip 3.0mu\left(s\right)$	$\mathcal{S}(f;s)$

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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Subsection 2.1(iii)

A short paragraph dealing with asymptotic approximations that are expressed in terms of two or more Poincaré asymptotic expansions has been added below (2.1.16).

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Equation (36.10.14)

36.10.14

3\left(\frac{{\partial}^{2}\Psi^{(\mathrm{E})}}{{\partial x}^{2}}-\frac{{% \partial}^{2}\Psi^{(\mathrm{E})}}{{\partial y}^{2}}\right)+2\mathrm{i}z\frac{% \partial\Psi^{(\mathrm{E})}}{\partial x}-x\Psi^{(\mathrm{E})}=0

Originally this equation appeared with $\frac{\partial\Psi^{(\mathrm{H})}}{\partial x}$ in the second term, rather than $\frac{\partial\Psi^{(\mathrm{E})}}{\partial x}$ .

Reported 2010-04-02.

…

38: 15.9 Relations to Other Functions

… ►The Jacobi transform is defined as …with inverse … … ►Any hypergeometric function for which a quadratic transformation exists can be expressed in terms of associated Legendre functions or Ferrers functions. … ►

§15.9(v) Complete Elliptic Integrals

…

39: 9.12 Scorer Functions

… ►where … ►

§9.12(vii) Integral Representations

… ►

Mellin–Barnes Type Integral

… ► … ►

Integrals

…

40: Bibliography K

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G. A. Kalugin, D. J. Jeffrey, and R. M. Corless (2012) Bernstein, Pick, Poisson and related integral expressions for Lambert

W

. Integral Transforms Spec. Funct. 23 (11), pp. 817–829.

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External Links:: ISSN 1065-2469, Document, Link, MathReview (Ross C. McPhedran), ZentralBlatt
Referenced by:: §4.13.
Encodings:: BibTeX

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Y. S. Kim, A. K. Rathie, and R. B. Paris (2013) An extension of Saalschütz’s summation theorem for the series

{}_{r+3}F_{r+2}

. Integral Transforms Spec. Funct. 24 (11), pp. 916–921.

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External Links:: ISSN 1065-2469, Document, FullText, MathReview (Christian Lavault), ZentralBlatt
Referenced by:: §16.4(ii), §16.4(ii).
Encodings:: BibTeX

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G. A. Kolesnik (1969) An improvement of the remainder term in the divisor problem. Mat. Zametki 6, pp. 545–554 (Russian).

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Notes:: In Russian. English translation: Math. Notes 6(1969), no. 5, pp. 784–791.
External Links:: Document, MathReview (A. I. Vinogradov), ZentralBlatt
Referenced by:: §27.11, Erratum (V1.1.8) for Section 27.11.
Encodings:: BibTeX

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T. H. Koornwinder and I. Sprinkhuizen-Kuyper (1978) Hypergeometric functions of

2\times 2

matrix argument are expressible in terms of Appel’s functions

F_{4}

. Proc. Amer. Math. Soc. 70 (1), pp. 39–42.

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External Links:: FullText, MathReview (P. N. Rathie), ZentralBlatt
Referenced by:: §35.7(ii).
Encodings:: BibTeX

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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.

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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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