Weber transform

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1: 10.22 Integrals

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10.22.44 $${\int }_0^{\mathrm{\infty }}{t}^{\mu }{Y}_{\nu }\left(t\right)dt=\frac{2^{\mu }}{\pi }\mathrm{\Gamma }\left(\frac{1}{2}\mu +\frac{1}{2}\nu +\frac{1}{2}\right)\mathrm{\Gamma }\left(\frac{1}{2}\mu -\frac{1}{2}\nu +\frac{1}{2}\right)\sin \left(\frac{1}{2}\mu -\frac{1}{2}\nu \right)\pi ,$$ $\mathrm{\Re }\left(\mu \pm \nu \right)>-1$, .

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Symbols:

$Y_{\nu }\left(z\right)$: Bessel function of the second kind, $\mathrm{\Gamma }\left(z\right)$: gamma function, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\int$: integral, $\mathrm{\Re }$: real part, $\sin z$: sine function and $\nu$: complex parameter

Keywords:

Mellin transform

A&S Ref:

11.4.19 (modified)

Referenced by:

§10.22(iii)

Permalink:

http://dlmf.nist.gov/10.22.E44

Encodings:

pMML, png, TeX

See also:

Annotations for §10.22(iv), §10.22 and Ch.10

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10.22.47 $${\int }_0^{\mathrm{\infty }}\frac{t^{\nu }{Y}_{\nu }\left(at\right)}{t^2+b^2}dt=-b^{\nu -1}{K}_{\nu }\left(ab\right),$$ .

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Symbols:

$Y_{\nu }\left(z\right)$: Bessel function of the second kind, $dx$: differential of $x$ , $\int$: integral, $K_{\nu }\left(z\right)$: modified Bessel function of the second kind, $\mathrm{\Re }$: real part and $\nu$: complex parameter

Keywords:

Mellin transform

A&S Ref:

11.4.46 (is the special case $\nu =0$.)

Referenced by:

§10.22(iv)

Permalink:

http://dlmf.nist.gov/10.22.E47

Encodings:

pMML, png, TeX

See also:

Annotations for §10.22(iv), §10.22 and Ch.10

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10.22.75

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Symbols:

$J_{\nu }\left(z\right)$: Bessel function of the first kind, $Y_{\nu }\left(z\right)$: Bessel function of the second kind, $\mathrm{\Gamma }\left(z\right)$: gamma function, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\int$: integral, $\nu$: complex parameter and $A$: area

Keywords:

Mellin transform

Referenced by:

§10.22(iv), §10.22(iv)

Permalink:

http://dlmf.nist.gov/10.22.E75

Encodings:

pMML, png, TeX

See also:

Annotations for §10.22(iv), §10.22(iv), §10.22 and Ch.10

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10.22.78 $$f(x)={\int }_0^{\mathrm{\infty }}{(xt)}^{\frac{1}{2}}\frac{J_{\nu }\left(xt\right)Y_{\nu }\left(at\right)-Y_{\nu }\left(xt\right)J_{\nu }\left(at\right)}{J_{\nu }^2\left(at\right)+Y_{\nu }^2\left(at\right)}{\int }_a^{\mathrm{\infty }}{(yt)}^{\frac{1}{2}}\left(J_{\nu }\left(yt\right)Y_{\nu }\left(at\right)-Y_{\nu }\left(yt\right)J_{\nu }\left(at\right)\right)f(y)dydt,$$ $a>0$.

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Symbols:

$J_{\nu }\left(z\right)$: Bessel function of the first kind, $Y_{\nu }\left(z\right)$: Bessel function of the second kind, $dx$: differential of $x$ , $\int$: integral, $x$: real variable, $y$: real variable, $\nu$: complex parameter and $f(x)$: function

Sources:

Titchmarsh (1962a, p. 87); Watson (1944, §14.52)

Referenced by:

§1.18(ix), §1.18(vi), §10.22(vi), Erratum (V1.2.0) for Chapter 10 Additions

Permalink:

http://dlmf.nist.gov/10.22.E78

Encodings:

pMML, png, TeX

Addition (effective with 1.2.0):

This equation was added.

See also:

Annotations for §10.22(v), §10.22 and Ch.10

►This is the Weber transform. …

2: 1.18 Linear Second Order Differential Operators and Eigenfunction Expansions

… ►For generalizations see the Weber transform (10.22.78) and an extended Bessel transform (10.22.79). …

3: 10.1 Special Notation

… ►For older notations see British Association for the Advancement of Science (1937, pp. xix–xx) and Watson (1944, Chapters 1–3).

4: 2.8 Differential Equations with a Parameter

… ►First we apply the Liouville transformation (§1.13(iv)) to (2.8.1). … ►The transformed equation has the form … ►The transformed differential equation is … ►The transformed differential equation is … ►For $J_{\nu }$ and $Y_{\nu }$ see §10.2(ii). …

5: Bibliography N

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On an integral transform involving a class of Mathieu functions. SIAM J. Math. Anal. 20 (6), pp. 1500–1513.

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External Links:

ISSN 0036-1410, Document, MathReview (K. C. Gupta), ZentralBlatt

Referenced by:

§28.26(ii).

Encodings:

BibTeX

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On an asymptotic expansion of the Kontorovich-Lebedev transform. Applicable Anal. 39 (4), pp. 249–263.

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External Links:

ISSN 0003-6811, Document, MathReview (Aloknath Chakrabarti), ZentralBlatt

Referenced by:

§10.43(v).

Encodings:

BibTeX

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On an asymptotic expansion of the Kontorovich-Lebedev transform. Methods Appl. Anal. 3 (1), pp. 98–108.

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External Links:

ISSN 1073-2772, FullText, MathReview (C. M. Joshi), ZentralBlatt

Referenced by:

§10.43(v).

Encodings:

BibTeX

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The resurgence properties of the large order asymptotics of the Anger-Weber function I. J. Class. Anal. 4 (1), pp. 1–39.

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External Links:

ISSN 1848-5979, Document, Link, MathReview (José Luis López), ZentralBlatt

Referenced by:

(11.11.10), (11.11.8), §11.11(iii).

Encodings:

BibTeX

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The resurgence properties of the large order asymptotics of the Anger-Weber function II. J. Class. Anal. 4 (2), pp. 121–147.

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External Links:

ISSN 1848-5979, Document, Link, MathReview Entry, ZentralBlatt

Referenced by:

(11.11.10), (11.11.8), §11.11(iii).

Encodings:

BibTeX

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6: Guide to Searching the DLMF

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Table 1: Query Examples

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Query	Matching records contain
"Fourier transform" and series	both the phrase “Fourier transform” and the word “series”.
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Fourier (transform or series)	at least one of “Fourier transform” or “Fourier series”.
1/(2pi) and "Fourier transform"	both $1/(2\pi )$ and the phrase “Fourier transform”.
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BesselJ_nu and BesselY_nu	both the Bessel functions $J_{\nu }$ and $Y_{\nu }$.
…

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7: Bibliography S

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The Laplace Transform: Theory and Applications. Undergraduate Texts in Mathematics, Springer-Verlag, New York.

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External Links:

ISBN 0-387-98698-7, MathReview (Philip Heywood), ZentralBlatt

Referenced by:

§1.14(iii), §1.14(vii).

Encodings:

BibTeX

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Numerical evaluation of the Hankel transform. Comput. Phys. Comm. 116 (2-3), pp. 278–294.

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External Links:

ISSN 0010-4655, Document, MathReview Entry, ZentralBlatt

Referenced by:

§10.74(vii).

Encodings:

BibTeX

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On integral representation of Weber’s parabolic cylinder function and its expansion into an infinite series. J. Indian Math. Soc. (N. S.) 4, pp. 34–38.

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External Links:

MathReview (M. C. Gray), ZentralBlatt

Referenced by:

§12.13(ii).

Encodings:

BibTeX

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On certain integrals and expansions involving Weber’s parabolic cylinder functions. J. Indian Math. Soc. (N. S.) 4, pp. 158–166.

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External Links:

MathReview (M. C. Gray), ZentralBlatt

Referenced by:

§12.13(ii).

Encodings:

BibTeX

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Numerical evaluation of spherical Bessel transforms via fast Fourier transforms. J. Comput. Phys. 100 (2), pp. 294–296.

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External Links:

ISSN 0021-9991, Document, MathReview, ZentralBlatt

Referenced by:

§10.74(vii).

Encodings:

BibTeX

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8: Bibliography V

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An integral transform involving Heun functions and a related eigenvalue problem. SIAM J. Math. Anal. 17 (3), pp. 688–703.

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External Links:

ISSN 0036-1410, Document, MathReview (William L. Perry), ZentralBlatt

Referenced by:

§31.10(i).

Encodings:

BibTeX

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Computational Frameworks for the Fast Fourier Transform. Frontiers in Applied Mathematics, Vol. 10, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA.

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External Links:

ISBN 0-89871-285-8, MathReview (S. Hitotumatu), ZentralBlatt

Referenced by:

§3.11(v).

Encodings:

BibTeX

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An infinite series of Weber’s parabolic cylinder functions. Proc. Benares Math. Soc. (N.S.) 3, pp. 37.

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External Links:

MathReview (M. C. Gray), ZentralBlatt

Referenced by:

§12.13(ii).

Encodings:

BibTeX

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Transformations of some Gauss hypergeometric functions. J. Comput. Appl. Math. 178 (1-2), pp. 473–487.

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External Links:

ISSN 0377-0427, Document, Link, MathReview (Wolfgang Bühring), ZentralBlatt

Referenced by:

§15.8(v).

Encodings:

BibTeX

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Representation of Lie Groups and Special Functions. Volume 1: Simplest Lie Groups, Special Functions and Integral Transforms. Mathematics and its Applications (Soviet Series), Vol. 72, Kluwer Academic Publishers Group, Dordrecht.

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Notes:

Translated from the Russian by V. A. Groza and A. A. Groza

External Links:

ISBN 0-7923-1466-2, MathReview (Robert A. Gustafson), ZentralBlatt

Referenced by:

§18.38(iii).

Encodings:

BibTeX

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9: Bibliography Z

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Integrali di funzioni di Anger, Weber ed Airy-Hardy. Rend. Sem. Mat. Univ. Padova 58, pp. 275–285 (Italian).

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External Links:

FullText, MathReview (M. E. Muldoon), ZentralBlatt

Referenced by:

§11.10(x).

Encodings:

BibTeX

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Distribution Theory and Transform Analysis, An Introduction and Generalized Functions with Applications. Dover, New York.

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Notes:

Reprint of 1965 McGraw-Hill Publication

External Links:

ISBN 0-486-65479-6, MathReview Entry

Referenced by:

§1.16(ix).

Encodings:

BibTeX

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Generalized Watson Transforms and Applications to Group Representations. Ph.D. Thesis, University of Vermont, Burlington,VT.

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Referenced by:

§35.7(ii).

Encodings:

BibTeX

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A fast Fourier-Bessel transform algorithm. Zh. Vychisl. Mat. i Mat. Fiz. 35 (7), pp. 1128–1133 (Russian).

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Notes:

English translation in Comput. Math. Math. Phys. 35(7), pp. 901–905

External Links:

ISSN 0044-4669, MathReview (Vítĕzslav Veselý), ZentralBlatt

Referenced by:

§10.74(vii).

Encodings:

BibTeX

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10: 13.8 Asymptotic Approximations for Large Parameters

… ►When the foregoing results are combined with Kummer’s transformation (13.2.39), an approximation is obtained for the case when $|b|$ is large, and $|b-a|$ and $|z|$ are bounded. … ►For the case $b>1$ the transformation (13.2.40) can be used. … ►

13.8.10 $$U(a,b,x)=\mathrm{\Gamma }\left(\frac{1}{2}b-a+\frac{1}{2}\right){\mathrm{e}}^{\frac{1}{2}x}{x}^{\frac{1}{2}-\frac{1}{2}b}\left(\cos \left(a\pi \right)J_{b-1}\left(\sqrt{2x(b-2a)}\right)-\sin \left(a\pi \right)Y_{b-1}\left(\sqrt{2x(b-2a)}\right)+\mathrm{env}{Y}_{b-1}\left(\sqrt{2x(b-2a)}\right)O\left({\left|a\right|}^{-\frac{1}{2}}\right)\right),$$

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Symbols:

$J_{\nu }\left(z\right)$: Bessel function of the first kind, $Y_{\nu }\left(z\right)$: Bessel function of the second kind, $O\left(x\right)$: order not exceeding, $\mathrm{\Gamma }\left(z\right)$: gamma function, $U(a,b,z)$: Kummer confluent hypergeometric function, $\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $\mathrm{env}{Y}_{\nu }\left(x\right)$: envelope of Bessel function $Y_{\nu }\left(x\right)$ , $\mathrm{e}$: base of natural logarithm, $\sin z$: sine function and $x$: real variable

A&S Ref:

13.5.15 (in different form)

Referenced by:

§13.9(ii)

Permalink:

http://dlmf.nist.gov/13.8.E10

Encodings:

pMML, png, TeX

See also:

Annotations for §13.8(iii), §13.8 and Ch.13

… ►where $C_{\nu }(a,\zeta )=\cos \left(\pi a\right)J_{\nu }\left(\zeta \right)+\sin \left(\pi a\right)Y_{\nu }\left(\zeta \right)$ and …