Bessel transform

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1: 10.74 Methods of Computation

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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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2: 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).

3: Bibliography Z

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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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4: Bibliography L

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Optimal cylindrical and spherical Bessel transforms satisfying bound state boundary conditions. Comput. Phys. Comm. 99 (2-3), pp. 297–306.

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Notes:

Single-precision Fortran, maximum accuracy 8S.

External Links:

ISSN 0010-4655, Document, Code, ZentralBlatt

Referenced by:

4th item.

Encodings:

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Bessel transforms and rational extrapolation. Numer. Math. 47 (1), pp. 1–14.

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External Links:

ISSN 0029-599X, Document, MathReview (Vítĕzslav Veselý), ZentralBlatt

Referenced by:

§10.74(vii).

Encodings:

BibTeX

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5: Bibliography O

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Tables of Bessel Transforms. Springer-Verlag, Berlin-New York.

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Notes:

Table errata: Math. Comp. v. 65 (1996), no. 215, p. 1386.

External Links:

ISBN 0-387-05997-0, MathReview, ZentralBlatt

Referenced by:

§1.14(viii), §10.22(v), §10.43(v), §10.43(vi), §11.10(x), §11.5(iii), §11.7(v), §11.9(iv), §13.10(v), §13.23(iii), §15.14, §18.17(ix), §8.14, §8.6(iii).

Encodings:

BibTeX

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6: Bibliography T

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LSFBTR: A subroutine for calculating spherical Bessel transforms. Comput. Phys. Comm. 30 (1), pp. 93–99.

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Notes:

Single-precision Fortran.

External Links:

ISSN 0010-4655, Document, Code

Referenced by:

9th item.

Encodings:

BibTeX

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7: 10.46 Generalized and Incomplete Bessel Functions; Mittag-Leffler Function

… ►The Laplace transform of $\varphi (\rho ,\beta ;z)$ can be expressed in terms of the Mittag-Leffler function: …

8: Bibliography S

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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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Computation of infinite integrals involving Bessel functions of arbitrary order by the $\overline{D}$-transformation. J. Comput. Appl. Math. 78 (1), pp. 125–130.

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External Links:

ISSN 0377-0427, Document, MathReview, ZentralBlatt

Referenced by:

§10.74(vii).

Encodings:

BibTeX

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On numerical Bessel transformation. Comput. Phys. Comm. 16 (3), pp. 383–387.

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Notes:

Single-precision Fortran, maximum accuracy 15D.

External Links:

ISSN 0010-4655, Document, Code

Referenced by:

8th item.

Encodings:

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9: 37.11 Spherical Harmonics

… ►See Braaksma and Meulenbeld (1968). … ►

37.11.30 $$ℱ\left(f\right)\left(\rho \xi \right)={\mathrm{i}}^{n}Y(\xi ){(2\pi \rho )}^{1-\frac{1}{2}d}{\int }_0^{\mathrm{\infty }}{f}_0(r)J_{\frac{1}{2}d+n-1}\left(\rho r\right)r^{\frac{1}{2}d}dr,$$ $\rho >0$, $\xi \in {𝕊}^{d-1}$,

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Symbols:

$J_{\nu }\left(z\right)$: Bessel function of the first kind, $ℱ\left(f\right)\left(s\right)$: Fourier transform, $dx$: differential of $x$ , $\in$: element of, $\mathrm{i}$: imaginary unit, $\int$: integral, $n$: nonnegative integer, $d$: positive integer, usually $\ge 2$ and ${𝕊}^{d-1}$: unit sphere of dimension $d-1$

Proved:

Stein and Weiss (1971, Ch. IV, Theorem 3.10)(proved)

Referenced by:

§37.11(iv)

Permalink:

http://dlmf.nist.gov/37.11.E30

Encodings:

pMML, png, TeX

See also:

Annotations for §37.11(iv), §37.11(iv), §37.11, Part 37.III and Ch.37

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37.11.31 $$ℱ\left(f\right)\left(\rho \cos \theta ,\rho \sin \theta )\right)={\mathrm{i}}^n{e}_n(\theta ){\int }_0^{\mathrm{\infty }}{f}_0(r)J_n\left(\rho r\right)rdr,$$ $\rho >0$.

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Symbols:

$J_{\nu }\left(z\right)$: Bessel function of the first kind, $ℱ\left(f\right)\left(s\right)$: Fourier transform, $\cos z$: cosine function, $dx$: differential of $x$ , $\mathrm{i}$: imaginary unit, $\int$: integral, $\sin z$: sine function and $n$: nonnegative integer

Proved:

Stein and Weiss (1971, Ch. IV, Theorem 1.6)(proved)

Referenced by:

(37.4.22)

Permalink:

http://dlmf.nist.gov/37.11.E31

Encodings:

pMML, png, TeX

See also:

Annotations for §37.11(iv), §37.11(iv), §37.11, Part 37.III and Ch.37

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10: 13.10 Integrals

… ►For additional Hankel transforms and also other Bessel transforms see Erdélyi et al. (1954b, §8.18) and Oberhettinger (1972, §§1.16 and 3.4.42–46, 4.4.45–47, 5.94–97). …