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21: 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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22: 10.76 Approximations

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Spherical Bessel Functions

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23: 4.42 Solution of Triangles

§4.42 Solution of Triangles

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§4.42(iii) Spherical Triangles

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Figure 4.42.3: Spherical triangle. Magnify

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24: 10.75 Tables

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

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The main tables in Abramowitz and Stegun (1964, Chapter 10) give ${𝗃}_n\left(x\right)$, ${𝗒}_n\left(x\right)$ $n=0(1)8$, $x=0(.1)10$, 5–8S; ${𝗃}_n\left(x\right)$, ${𝗒}_n\left(x\right)$ $n=0(1)20(10)50$, 100, $x=1,2,5,10,50,100$, 10S; ${𝗂}_n^{(1)}\left(x\right)$, ${𝗄}_n\left(x\right)$, $n=0,1,2$, $x=0(.1)5$, 4–9D; ${𝗂}_n^{(1)}\left(x\right)$, ${𝗄}_n\left(x\right)$, $n=0(1)20(10)50$, 100, $x=1,2,5,10,50,100$, 10S. (For the notation see §10.1 and §10.47(ii).)

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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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Olver (1960) tabulates $a_{n,m}^{\prime }$, ${𝗃}_n\left(a_{n,m}^{\prime }\right)$, $b_{n,m}^{\prime }$, ${𝗒}_n\left(b_{n,m}^{\prime }\right)$, $n=1(1)20$, $m=1(1)50$, 8D. Also included are tables of the coefficients in the uniform asymptotic expansions of these zeros and associated values as $n\to \mathrm{\infty }$.

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25: 10.74 Methods of Computation

… ►Similar observations apply to the computation of modified Bessel functions, spherical Bessel functions, and Kelvin functions. … ►Similar considerations apply to the spherical Bessel functions and Kelvin functions. … ► … ►

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. …

26: 34.3 Basic Properties: $\mathit{3}j$ Symbol

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§34.3(vii) Relations to Legendre Polynomials and Spherical Harmonics

►For the polynomials $P_l$ see §18.3, and for the function $Y_{l,m}$ see §14.30. ►

34.3.19

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Symbols:

$P_n\left(x\right)$: Legendre polynomial, $\cos z$: cosine function, : $\mathit{3}j$ symbol and $l,l_r$: non-negative integers or non-negative integers plus one half.

Referenced by:

§14.18(iv), §14.30(iii), §34.3(vii)

Permalink:

http://dlmf.nist.gov/34.3.E19

Encodings:

pMML, png, TeX

See also:

Annotations for §34.3(vii), §34.3 and Ch.34

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34.3.20

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\overline{z}$: complex conjugate, $Y_{l,m}(\theta ,\varphi )$: spherical harmonic, : $\mathit{3}j$ symbol and $l,l_r$: non-negative integers or non-negative integers plus one half.

Permalink:

http://dlmf.nist.gov/34.3.E20

Encodings:

pMML, png, TeX

Notational Change (effective with 1.0.19):

As a notational clarification, the spherical harmonic in the sum over $l,m$ has been explicity marked up as a complex conjugate.

See also:

Annotations for §34.3(vii), §34.3 and Ch.34

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34.3.22

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\int$: integral, $\sin z$: sine function, $Y_{l,m}(\theta ,\varphi )$: spherical harmonic, : $\mathit{3}j$ symbol and $l,l_r$: non-negative integers or non-negative integers plus one half.

Referenced by:

§14.30(ii), §34.3(vii)

Permalink:

http://dlmf.nist.gov/34.3.E22

Encodings:

pMML, png, TeX

See also:

Annotations for §34.3(vii), §34.3 and Ch.34

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27: 15.17 Mathematical Applications

… ►First, as spherical functions on noncompact Riemannian symmetric spaces of rank one, but also as associated spherical functions, intertwining functions, matrix elements of SL$(2,ℝ)$, and spherical functions on certain nonsymmetric Gelfand pairs. …

28: 18.41 Tables

… ►For $P_n\left(x\right)$ ($={𝖯}_n\left(x\right)$) see §14.33. … ►For $P_n\left(x\right)$, $L_n\left(x\right)$, and $H_n\left(x\right)$ see §3.5(v). …

29: 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 …

30: 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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1.17.22 $$\delta \left(x-a\right)=\sum _{k=0}^{\mathrm{\infty }}(k+\frac{1}{2})P_k\left(x\right)P_k\left(a\right).$$

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Symbols:

$\delta \left(x-a\right)$: Dirac delta (or Dirac delta function), $P_n\left(x\right)$: Legendre polynomial and $k$: integer

Referenced by:

§1.17(iii), §1.17(iv), §14.18(iii)

Permalink:

http://dlmf.nist.gov/1.17.E22

Encodings:

pMML, png, TeX

See also:

Annotations for §1.17(iii), §1.17(iii), §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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