spherical Bessel functions

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21: 33.9 Expansions in Series of Bessel Functions

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§33.9(i) Spherical Bessel Functions

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33.9.1 $$F_{\mathrm{\ell }}(\eta ,\rho )=\rho \sum _{k=0}^{\mathrm{\infty }}{a}_k{𝗃}_{\mathrm{\ell }+k}\left(\rho \right),$$

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Symbols:

$F_{\mathrm{\ell }}(\eta ,\rho )$: regular Coulomb radial function, ${𝗃}_n\left(z\right)$: spherical Bessel function of the first kind, $k$: nonnegative integer, $\mathrm{\ell }$: nonnegative integer, $\rho$: nonnegative real variable, $\eta$: real parameter and $a_k$: coefficients

A&S Ref:

14.4.5

Referenced by:

§33.9(i), §33.9(i)

Permalink:

http://dlmf.nist.gov/33.9.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §33.9(i), §33.9 and Ch.33

►where the function $𝗃$ is as in §10.47(ii), $a_{-1}=0$, $a_0=(2\mathrm{\ell }+1)!!C_{\mathrm{\ell }}\left(\eta \right)$, and …

22: 10.59 Integrals

§10.59 Integrals

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10.59.1

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Symbols:

$P_n\left(x\right)$: Legendre polynomial, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $\int$: integral, ${𝗃}_n\left(z\right)$: spherical Bessel function of the first kind and $n$: integer

A&S Ref:

11.4.26

Referenced by:

§10.59

Permalink:

http://dlmf.nist.gov/10.59.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §10.59 and Ch.10

… ►For an integral representation of the Dirac delta in terms of a product of spherical Bessel functions of the first kind see §1.17(ii), and for a generalization see Maximon (1991). …

23: 8.7 Series Expansions

§8.7 Series Expansions

►For the functions $e_n(z)$, ${𝗂}_n^{(1)}\left(z\right)$, and $L_n^{(\alpha )}\left(x\right)$ see (8.4.11), §§10.47(ii), and 18.3, respectively. … ►

8.7.5 $${\gamma }^{\ast }(a,z)={\mathrm{e}}^{-\frac{1}{2}z}\sum _{n=0}^{\mathrm{\infty }}\frac{{\left(1-a\right)}_n}{\mathrm{\Gamma }\left(n+a+1\right)}\left(2n+1\right){𝗂}_n^{(1)}\left(\frac{1}{2}z\right).$$

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, ${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $\mathrm{e}$: base of natural logarithm, ${\gamma }^{\ast }(a,z)$: incomplete gamma function, ${𝗂}_n^{(1)}\left(z\right)$: modified spherical Bessel function, $z$: complex variable, $a$: parameter and $n$: nonnegative integer

Referenced by:

§8.7

Permalink:

http://dlmf.nist.gov/8.7.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §8.7 and Ch.8

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24: 33.5 Limiting Forms for Small $\rho$, Small $|\eta |$, or Large $\mathrm{\ell }$

… ►For the functions $𝗃$, $𝗒$, $J$, $Y$ see §§10.47(ii), 10.2(ii). …

25: 10.77 Software

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§10.77(ii) Bessel Functions–Real Argument and Integer or Half-Integer Order (including Spherical Bessel Functions)

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26: 10.75 Tables

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§10.75(ix) Spherical Bessel Functions, Modified Spherical Bessel Functions, and their Derivatives

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Zhang and Jin (1996, pp. 296–305) tabulates ${𝗃}_n\left(x\right)$, ${𝗃}_n^{\prime }\left(x\right)$, ${𝗒}_n\left(x\right)$, ${𝗒}_n^{\prime }\left(x\right)$, ${𝗂}_n^{(1)}\left(x\right)$, $𝗂_n^{(1)}{}_{}{}^{\prime }\left(x\right)$, ${𝗄}_n\left(x\right)$, ${𝗄}_n^{\prime }\left(x\right)$, $n=0(1)10(10)30$, 50, 100, $x=1$, 5, 10, 25, 50, 100, 8S; $x{𝗃}_n\left(x\right)$, ${(x{𝗃}_n\left(x\right))}^{\prime }$, $x{𝗒}_n\left(x\right)$, ${(x{𝗒}_n\left(x\right))}^{\prime }$ (Riccati–Bessel functions and their derivatives), $n=0(1)10(10)30$, 50, 100, $x=1$, 5, 10, 25, 50, 100, 8S; real and imaginary parts of ${𝗃}_n\left(z\right)$, ${𝗃}_n^{\prime }\left(z\right)$, ${𝗒}_n\left(z\right)$, ${𝗒}_n^{\prime }\left(z\right)$, ${𝗂}_n^{(1)}\left(z\right)$, $𝗂_n^{(1)}{}_{}{}^{\prime }\left(z\right)$, ${𝗄}_n\left(z\right)$, ${𝗄}_n^{\prime }\left(z\right)$, $n=0(1)15$, 20(10)50, 100, $z=4+2\mathrm{i}$, $20+10\mathrm{i}$, 8S. (For the notation replace $j,y,\mathrm{i},k$ by $𝗃$, $𝗒$, ${𝗂}^{(1)}$, $𝗄$, respectively.)

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§10.75(x) Zeros and Associated Values of Derivatives of Spherical Bessel Functions

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27: 1.17 Integral and Series Representations of the Dirac Delta

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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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28: 6.18 Methods of Computation

… ►For small or moderate values of $x$ and $|z|$, the expansion in power series (§6.6) or in series of spherical Bessel functions (§6.10(ii)) can be used. …However, this problem is less severe for the series of spherical Bessel functions because of their more rapid rate of convergence, and also (except in the case of (6.10.6)) absence of cancellation when $z=x$ ($>0$). …

29: 30.2 Differential Equations

… ►If $\gamma =0$, Equation (30.2.4) is satisfied by spherical Bessel functions; see (10.47.1).

30: 8.21 Generalized Sine and Cosine Integrals

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Spherical-Bessel-Function Expansions

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8.21.16 $$\mathrm{Si}(a,z)=z^a\sum _{k=0}^{\mathrm{\infty }}\frac{\left(2k+\frac{3}{2}\right){\left(1-\frac{1}{2}a\right)}_k}{{\left(\frac{1}{2}+\frac{1}{2}a\right)}_{k+1}}{𝗃}_{2k+1}\left(z\right),$$ $a\ne -1,-3,-5,\mathrm{\dots }$,

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Symbols:

${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $\mathrm{Si}(a,z)$: generalized sine integral, ${𝗃}_n\left(z\right)$: spherical Bessel function of the first kind, $z$: complex variable, $a$: parameter and $k$: nonnegative integer

Referenced by:

§8.21(vi)

Permalink:

http://dlmf.nist.gov/8.21.E16

Encodings:

pMML, png, TeX

See also:

Annotations for §8.21(vi), §8.21(vi), §8.21 and Ch.8

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8.21.17 $$\mathrm{Ci}(a,z)=z^a\sum _{k=0}^{\mathrm{\infty }}\frac{\left(2k+\frac{1}{2}\right){\left(\frac{1}{2}-\frac{1}{2}a\right)}_k}{{\left(\frac{1}{2}a\right)}_{k+1}}{𝗃}_{2k}\left(z\right),$$ $a\ne 0,-2,-4,\mathrm{\dots }$.

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Symbols:

${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $\mathrm{Ci}(a,z)$: generalized cosine integral, ${𝗃}_n\left(z\right)$: spherical Bessel function of the first kind, $z$: complex variable, $a$: parameter and $k$: nonnegative integer

Referenced by:

§8.21(vi)

Permalink:

http://dlmf.nist.gov/8.21.E17

Encodings:

pMML, png, TeX

See also:

Annotations for §8.21(vi), §8.21(vi), §8.21 and Ch.8

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