spherical 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. …

2: 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:

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3: 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:

BibTeX

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4: 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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5: 1.14 Integral Transforms

§1.14 Integral Transforms

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6: 37.11 Spherical Harmonics

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

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

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Defines:

$\rho$: non-negative real parameter (locally)

Symbols:

$J_{\nu }\left(z\right)$: Bessel function of the first kind, $ℱ\left(\varphi \right)\left(𝐱\right)$: Fourier transform of a function on Euclidean space, $dx$: differential of $x$ , $\in$: element of, $\mathrm{i}$: imaginary unit, $\int$: integral, $n$: nonnegative integer, $d$: positive integer, usually $\ge 2$ , $f$: smooth function of $d$-variables, $𝝃$: $d$-dimensional vector restricted to ${𝕊}^{d-1}$ , ${𝕊}^d$: unit sphere of dimension $d$ , $Y$: spherical harmonic of degree $n$ of $(d-1)$-variables and $r$: non-negative real parameter

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:

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See also:

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

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7: 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).

8: 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. … ►

Bessel Functions and Spherical Bessel Functions (§§10.2(ii), 10.47(ii))

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1.17.14 $$\delta \left(x-a\right)=\frac{2xa}{\pi }{\int }_0^{\mathrm{\infty }}{t}^2{𝗃}_{\mathrm{\ell }}\left(xt\right){𝗃}_{\mathrm{\ell }}\left(at\right)dt,$$ $x>0$, $a>0$.

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Symbols:

$\delta \left(x-a\right)$: Dirac delta (or Dirac delta function), $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\int$: integral and ${𝗃}_n\left(z\right)$: spherical Bessel function of the first kind

Referenced by:

§1.17(ii), §1.18(vi)

Permalink:

http://dlmf.nist.gov/1.17.E14

Encodings:

pMML, png, TeX

See also:

Annotations for §1.17(ii), §1.17(ii), §1.17 and Ch.1

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Spherical Harmonics (§14.30)

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1.17.25 $$\delta \left(\cos {\theta }_1-\cos {\theta }_2\right)\delta \left({\varphi }_1-{\varphi }_2\right)=\sum _{\mathrm{\ell }=0}^{\mathrm{\infty }}\sum _{m=-\mathrm{\ell }}^{\mathrm{\ell }}{Y}_{\mathrm{\ell },m}({\theta }_1,{\varphi }_1)\overline{Y_{\mathrm{\ell },m}({\theta }_2,{\varphi }_2)}.$$

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Symbols:

$\delta \left(x-a\right)$: Dirac delta (or Dirac delta function), $\overline{z}$: complex conjugate, $\cos z$: cosine function, $Y_{l,m}(\theta ,\varphi )$: spherical harmonic and $m$: nonnegative integer

Referenced by:

§1.17(iii)

Permalink:

http://dlmf.nist.gov/1.17.E25

Encodings:

pMML, png, TeX

Notational Change (effective with 1.0.19):

As a notational clarification, the second spherical harmonic in the sum over $\mathrm{\ell }$ has been explicity marked up as a complex conjugate.

See also:

Annotations for §1.17(iii), §1.17(iii), §1.17 and Ch.1

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9: 18.17 Integrals

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§18.17(v) Fourier Transforms

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Jacobi

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Ultraspherical

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§18.17(vi) Laplace Transforms

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§18.17(vii) Mellin Transforms

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10: Bibliography D

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Complex zeros of linear combinations of spherical Bessel functions and their derivatives. SIAM J. Math. Anal. 4 (1), pp. 128–133.

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External Links:

ISSN 1095-7154, Document, MathReview (C. J. Bouwkamp), ZentralBlatt

Referenced by:

§10.58.

Encodings:

BibTeX

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Integral Transforms and their Applications. 2nd edition, Applied Mathematical Sciences, Vol. 25, Springer-Verlag, New York.

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External Links:

ISBN 0-387-96080-5, MathReview, ZentralBlatt

Referenced by:

§1.14(vii).

Encodings:

BibTeX

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Integral transforms and their applications. Third edition, CRC Press, Boca Raton, FL.

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External Links:

ISBN 978-1-4822-2357-6, MathReview Entry, ZentralBlatt

Referenced by:

§1.16(viii).

Encodings:

BibTeX

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Chebyshev series for the spherical Bessel function $j_l(r)$ . Comput. Phys. Comm. 18 (1), pp. 73–86.

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External Links:

Document

Referenced by:

§10.76(iii).

Encodings:

BibTeX

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Handbuch der Laplace-Transformation. Bd. II. Anwendungen der Laplace-Transformation. 1. Abteilung. Birkhäuser Verlag, Basel und Stuttgart (German).

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External Links:

MathReview (I. I. Hirschman), ZentralBlatt

Referenced by:

§2.5(i).

Encodings:

BibTeX

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