Hankel inversion theorem

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1: 10.22 Integrals

… ►Hankel’s inversion theorem is given by ►

10.22.77 $$f(y)={\int }_0^{\mathrm{\infty }}g(x)J_{\nu }\left(xy\right){(xy)}^{\frac{1}{2}}dx.$$

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Symbols:

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

Referenced by:

§1.18(vi), §10.22(v)

Permalink:

http://dlmf.nist.gov/10.22.E77

Encodings:

pMML, png, TeX

See also:

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

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2: Bibliography C

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A quadrature formula for the Hankel transform. Numer. Algorithms 9 (2), pp. 343–354.

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External Links:

ISSN 1017-1398, Document, MathReview (V. I. Polovinkin), ZentralBlatt

Referenced by:

§10.74(vii).

Encodings:

BibTeX

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The Staudt-Clausen theorem. Math. Mag. 34, pp. 131–146.

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External Links:

MathReview (K. Goldberg), ZentralBlatt

Referenced by:

§24.6.

Encodings:

BibTeX

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Short proofs of three theorems on elliptic integrals. SIAM J. Math. Anal. 9 (3), pp. 524–528.

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External Links:

Document, MathReview (Y. L. Luke), ZentralBlatt

Referenced by:

§19.26(i).

Encodings:

BibTeX

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Optimized fast Hankel transform filters. Geophysical Prospecting 38 (5), pp. 545–568.

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External Links:

Document

Referenced by:

§10.74(vii).

Encodings:

BibTeX

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The universal von Staudt theorems. Trans. Amer. Math. Soc. 315 (2), pp. 591–603.

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External Links:

ISSN 0002-9947, FullText, MathReview (T. Metsänkylä), ZentralBlatt

Referenced by:

§24.16(iii).

Encodings:

BibTeX

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3: Bibliography R

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Normal limit theorems for symmetric random matrices. Probab. Theory Related Fields 112 (3), pp. 411–423.

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External Links:

ISSN 0178-8051, Document, MathReview (Alexei Khorunzhy), ZentralBlatt

Referenced by:

§35.9.

Encodings:

BibTeX

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Computation of Hankel (Bessel) functions of complex index and argument by numerical integration of a Schläfli contour integral. Ž. Vyčisl. Mat. i Mat. Fiz. 13, pp. 1415–1424, 1636.

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Notes:

English translation in U.S.S.R. Computational Math. and Math. Phys. 13(6), pp. 58–67

External Links:

ISSN 0044-4669, Document, MathReview (H. Zegeling), ZentralBlatt

Referenced by:

§10.74(iii).

Encodings:

BibTeX

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13 Lectures on Fermat’s Last Theorem. Springer-Verlag, New York.

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External Links:

ISBN 0-387-90432-8, MathReview (T. Metsänkylä), ZentralBlatt

Referenced by:

§24.10(iv), §24.17(iii).

Encodings:

BibTeX

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Finite-sum rules for Macdonald’s functions and Hankel’s symbols. Integral Transform. Spec. Funct. 10 (2), pp. 115–124.

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External Links:

ISSN 1065-2469, Document, MathReview, ZentralBlatt

Referenced by:

§10.60(iii).

Encodings:

BibTeX

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Dawson’s integral and the sampling theorem. Computers in Physics 3 (2), pp. 85–87.

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Referenced by:

§7.25(vi).

Encodings:

BibTeX

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