Weber transform

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11—19 of 19 matching pages

11: Bibliography M

… ►

J. P. McClure and R. Wong (1978) Explicit error terms for asymptotic expansions of Stieltjes transforms. J. Inst. Math. Appl. 22 (2), pp. 129–145.

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External Links:: ISSN 0020-2932, Document, MathReview (G. M. Habibullah), ZentralBlatt
Referenced by:: §2.6(ii).
Encodings:: BibTeX

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J. C. P. Miller (1952) On the choice of standard solutions to Weber’s equation. Proc. Cambridge Philos. Soc. 48, pp. 428–435.

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External Links:: Document, MathReview (W. Wasow), ZentralBlatt
Referenced by:: §12.1, §12.14(i), §12.2(i).
Encodings:: BibTeX

►

J. C. P. Miller (Ed.) (1955) Tables of Weber Parabolic Cylinder Functions. Her Majesty’s Stationery Office, London.

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External Links:: MathReview (A. Erdélyi), ZentralBlatt
Referenced by:: §10.16, §10.39, §12.14(i), §12.14(ii), §12.14(iv), §12.14(v), §12.14(v), §12.14(vi), §12.14(vii), §12.14(viii), §12.14(x), §12.14(x), 2nd item, §12.2(i), §12.2(ii), §12.2(iii), §12.2(iv), §12.2(v), §12.2(vi), §12.2(vi), §12.4, §12.5(i), §12.5(ii), §12.5(iii), §12.7(i), §12.7(ii), §12.7(iii), §12.7(iii), §12.7(iv), §12.8(i), Ch.12.
Encodings:: BibTeX

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A. E. Milne, P. A. Clarkson, and A. P. Bassom (1997) Bäcklund transformations and solution hierarchies for the third Painlevé equation. Stud. Appl. Math. 98 (2), pp. 139–194.

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External Links:: ISSN 0022-2526, Document, MathReview (Alexander Vladimirovich Kitaev), ZentralBlatt
Referenced by:: §32.10(iii), §32.7(iii), §32.8(iii), §32.9(i).
Encodings:: BibTeX

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S. C. Milne (1994) A

q

-analog of a Whipple’s transformation for hypergeometric series in

U(n)

. Adv. Math. 108 (1), pp. 1–76.

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External Links:: ISSN 0001-8708, Document, MathReview, ZentralBlatt
Referenced by:: §17.15.
Encodings:: BibTeX

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12: 10.9 Integral Representations

… ►

10.9.3

Y_{0}\left(z\right)=\frac{4}{\pi^{2}}\int_{0}^{\frac{1}{2}\pi}\cos\left(z\cos% \theta\right)\left(\gamma+\ln\left(2z{\sin}^{2}\theta\right)\right)\,\mathrm{d% }\theta,

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Symbols:: $Y_{\NVar{\nu}}\left(\NVar{z}\right)$ : Bessel function of the second kind, $\gamma$ : Euler’s constant, $\pi$ : the ratio of the circumference of a circle to its diameter, $\cos\NVar{z}$ : cosine function, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\int$ : integral, $\ln\NVar{z}$ : principal branch of logarithm function, $\sin\NVar{z}$ : sine function and $z$ : complex variable
A&S Ref:: 9.19
Referenced by:: §10.9(i)
Permalink:: http://dlmf.nist.gov/10.9.E3
Encodings:: pMML, png, TeX
See also:: Annotations for §10.9(i), §10.9(i), §10.9 and Ch.10

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10.9.7

Y_{\nu}\left(z\right)=\frac{1}{\pi}\int_{0}^{\pi}\sin\left(z\sin\theta-\nu% \theta\right)\,\mathrm{d}\theta-\frac{1}{\pi}\int_{0}^{\infty}\left(e^{\nu t}+% e^{-\nu t}\cos\left(\nu\pi\right)\right)e^{-z\sinh t}\,\mathrm{d}t,

|\operatorname{ph}z|<\tfrac{1}{2}\pi

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Symbols:: $Y_{\NVar{\nu}}\left(\NVar{z}\right)$ : Bessel function of the second kind, $\pi$ : the ratio of the circumference of a circle to its diameter, $\cos\NVar{z}$ : cosine function, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\mathrm{e}$ : base of natural logarithm, $\sinh\NVar{z}$ : hyperbolic sine function, $\int$ : integral, $\operatorname{ph}$ : phase, $\sin\NVar{z}$ : sine function, $z$ : complex variable and $\nu$ : complex parameter
A&S Ref:: 9.1.22
Permalink:: http://dlmf.nist.gov/10.9.E7
Encodings:: pMML, png, TeX
See also:: Annotations for §10.9(i), §10.9(i), §10.9 and Ch.10

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Y_{\nu}\left(x\right)=-\frac{2}{\pi}\int_{0}^{\infty}\cos\left(x\cosh t-\tfrac% {1}{2}\nu\pi\right)\cosh\left(\nu t\right)\,\mathrm{d}t,

|\Re\nu|<1,x>0

… ►

Y_{0}\left(x\right)=-\frac{2}{\pi}\int_{0}^{\infty}\cos\left(x\cosh t\right)\,% \mathrm{d}t,

x>0

… ►

10.9.30

{J_{\nu}}^{2}\left(z\right)+{Y_{\nu}}^{2}\left(z\right)=\frac{8}{\pi^{2}}\int_% {0}^{\infty}\cosh\left(2\nu t\right)K_{0}\left(2z\sinh t\right)\,\mathrm{d}t,

|\operatorname{ph}z|<\tfrac{1}{2}\pi

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Symbols:: $J_{\NVar{\nu}}\left(\NVar{z}\right)$ : Bessel function of the first kind, $Y_{\NVar{\nu}}\left(\NVar{z}\right)$ : Bessel function of the second kind, $\pi$ : the ratio of the circumference of a circle to its diameter, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\cosh\NVar{z}$ : hyperbolic cosine function, $\sinh\NVar{z}$ : hyperbolic sine function, $\int$ : integral, $K_{\NVar{\nu}}\left(\NVar{z}\right)$ : modified Bessel function of the second kind, $\operatorname{ph}$ : phase, $z$ : complex variable and $\nu$ : complex parameter
Permalink:: http://dlmf.nist.gov/10.9.E30
Encodings:: pMML, png, TeX
See also:: Annotations for §10.9(iii), §10.9(iii), §10.9 and Ch.10

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13: 1.17 Integral and Series Representations of the Dirac Delta

… ►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. … ►See Arfken and Weber (2005, Eq. (11.59)) and Konopinski (1981, p. 242). … ►For (1.17.25) see Arfken and Weber (2005, p. 792). …

14: Bibliography F

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J. Faraut (1982) Un théorème de Paley-Wiener pour la transformation de Fourier sur un espace riemannien symétrique de rang un. J. Funct. Anal. 49 (2), pp. 230–268.

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External Links:: ISSN 0022-1236, Document, Link, MathReview (Mogens Flensted-Jensen), ZentralBlatt
Referenced by:: §18.39(i).
Encodings:: BibTeX

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H. E. Fettis (1965) Calculation of elliptic integrals of the third kind by means of Gauss’ transformation. Math. Comp. 19 (89), pp. 97–104.

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External Links:: FullText, MathReview (J. E. L. Peck), ZentralBlatt
Referenced by:: §19.36(ii).
Encodings:: BibTeX

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F. Feuillebois (1991) Numerical calculation of singular integrals related to Hankel transform. Comput. Math. Appl. 21 (2-3), pp. 87–94.

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Notes:: Fortran, minimum precision 6D.
External Links:: ISSN 0898-1221, Document, MathReview, ZentralBlatt
Referenced by:: §10.77(ix).
Encodings:: BibTeX

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L. Fox (1960) Tables of Weber Parabolic Cylinder Functions and Other Functions for Large Arguments. National Physical Laboratory Mathematical Tables, Vol. 4. Department of Scientific and Industrial Research, Her Majesty’s Stationery Office, London.

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External Links:: MathReview (L. J. Slater), ZentralBlatt
Referenced by:: 3rd item, 2nd item.
Encodings:: BibTeX

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C. L. Frenzen (1987b) On the asymptotic expansion of Mellin transforms. SIAM J. Math. Anal. 18 (1), pp. 273–282.

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External Links:: ISSN 0036-1410, Document, Link, MathReview (Archibald Brown), ZentralBlatt
Referenced by:: §2.3(vi).
Encodings:: BibTeX

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15: 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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I. Ye. Kireyeva and K. A. Karpov (1961) Tables of Weber functions. Vol. I. Mathematical Tables Series, Vol. 15, Pergamon Press, London-New York.

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Notes:: Translated by Prasenjit Basu
External Links:: MathReview (F. W. J. Olver), ZentralBlatt
Referenced by:: 4th item.
Encodings:: BibTeX

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T. H. Koornwinder (1975a) A new proof of a Paley-Wiener type theorem for the Jacobi transform. Ark. Mat. 13, pp. 145–159.

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External Links:: Document, MathReview (G. Gasper), ZentralBlatt
Referenced by:: §18.10(i).
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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V. I. Krylov and N. S. Skoblya (1985) A Handbook of Methods of Approximate Fourier Transformation and Inversion of the Laplace Transformation. Mir, Moscow.

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Notes:: Translated from the Russian by George Yankovsky. Reprint of the 1977 translation
External Links:: MathReview, ZentralBlatt
Referenced by:: §3.5(viii).
Encodings:: BibTeX

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16: 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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G. E. Andrews (1972) Summations and transformations for basic Appell series. J. London Math. Soc. (2) 4, pp. 618–622.

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External Links:: MathReview (R. N. Jain), ZentralBlatt
Referenced by:: §17.11.
Encodings:: BibTeX

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G. B. Arfken and H. J. Weber (2005) Mathematical Methods for Physicists. 6th edition, Elsevier, Oxford.

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External Links:: ISBN 0-12-059876-0;0-12-088584-0, MathReview, ZentralBlatt
Referenced by:: §1.17(ii), §1.17(iii).
Encodings:: BibTeX

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17: 10.74 Methods of Computation

… ►Then

J_{n}\left(x\right)

and

Y_{n}\left(x\right)

can be generated by either forward or backward recurrence on

n

when

n<x

, but if

n>x

then to maintain stability

J_{n}\left(x\right)

has to be generated by backward recurrence on

n

, and

Y_{n}\left(x\right)

has to be generated by forward recurrence on

n

. … ►

Hankel Transform

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Spherical Bessel Transform

►The spherical Bessel transform is the Hankel transform (10.22.76) in the case when

\nu

is half an odd positive integer. … ►

Kontorovich–Lebedev Transform

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18: 32.10 Special Function Solutions

… ►Solutions for other values of

\alpha

are derived from

w(z;\pm\tfrac{1}{2})

by application of the Bäcklund transformations (32.7.1) and (32.7.2). … ►

32.10.14

\phi(z)=z^{\nu}\left(C_{1}J_{\nu}\left(\zeta\right)+C_{2}Y_{\nu}\left(\zeta% \right)\right),

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Symbols:: $J_{\NVar{\nu}}\left(\NVar{z}\right)$ : Bessel function of the first kind, $Y_{\NVar{\nu}}\left(\NVar{z}\right)$ : Bessel function of the second kind, $z$ : real, $\phi(z)$ : function, $\nu$ and $C_{j}$ : constants
Permalink:: http://dlmf.nist.gov/32.10.E14
Encodings:: pMML, png, TeX
See also:: Annotations for §32.10(iii), §32.10 and Ch.32

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19: Bibliography H

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P. I. Hadži (1973) The Laplace transform for expressions that contain a probability function. Bul. Akad. Štiince RSS Moldoven. 1973 (2), pp. 78–80, 93 (Russian).

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External Links:: MathReview (L. Berg)
Referenced by:: §7.14(iii).
Encodings:: BibTeX

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N. Hale and A. Townsend (2016) A fast FFT-based discrete Legendre transform. IMA J. Numer. Anal. 36 (4), pp. 1670–1684.

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External Links:: ISSN 0272-4979, Document, Link, MathReview (Denis N. Sidorov)
Referenced by:: §18.16(iii).
Encodings:: BibTeX

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R. A. Handelsman and J. S. Lew (1970) Asymptotic expansion of Laplace transforms near the origin. SIAM J. Math. Anal. 1 (1), pp. 118–130.

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External Links:: Document, MathReview (E. Riekstiņš), ZentralBlatt
Referenced by:: §2.5(iii), §2.5(ii).
Encodings:: BibTeX

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E. W. Hansen (1985) Fast Hankel transform algorithm. IEEE Trans. Acoust. Speech Signal Process. 32 (3), pp. 666–671.

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External Links:: ISSN 0096-3518, FullText, ZentralBlatt
Referenced by:: §10.74(vii).
Encodings:: BibTeX

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P. Hillion (1997) Diffraction and Weber functions. SIAM J. Appl. Math. 57 (6), pp. 1702–1715.

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External Links:: ISSN 0036-1399, Document, MathReview, ZentralBlatt
Referenced by:: §12.13(ii), §12.17.
Encodings:: BibTeX

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