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1: 14.30 Spherical and Spheroidal Harmonics

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§14.30(i) Definitions

… ► … ►As an example, Laplace’s equation

\nabla^{2}W=0

in spherical coordinates (§1.5(ii)): … ►Here, in spherical coordinates,

\mathrm{L}^{2}

is the squared angular momentum operator: … ►

2: 10.73 Physical Applications

… ►See Krivoshlykov (1994, Chapter 2, §2.2.10; Chapter 5, §5.2.2), Kapany and Burke (1972, Chapters 4–6; Chapter 7, §A.1), and Slater (1942, Chapter 4, §§20, 25). … ► … ►

§10.73(ii) Spherical Bessel Functions

►The functions

\mathsf{j}_{n}\left(x\right)

\mathsf{y}_{n}\left(x\right)

{\mathsf{h}^{(1)}_{n}}\left(x\right)

, and

{\mathsf{h}^{(2)}_{n}}\left(x\right)

arise in the solution (again by separation of variables) of the Helmholtz equation in spherical coordinates

\rho,\theta,\phi

(§1.5(ii)): …Accordingly, the spherical Bessel functions appear in all problems in three dimensions with spherical symmetry involving the scattering of electromagnetic radiation. …

3: 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 $\mathsf{j}_{n}\left(x\right)$ , $\mathsf{y}_{n}\left(x\right)$ $n=0(1)8$ , $x=0(.1)10$ , 5–8S; $\mathsf{j}_{n}\left(x\right)$ , $\mathsf{y}_{n}\left(x\right)$ $n=0(1)20(10)50$ , 100, $x=1,2,5,10,50,100$ , 10S; ${\mathsf{i}^{(1)}_{n}}\left(x\right)$ , $\mathsf{k}_{n}\left(x\right)$ , $n=0,1,2$ , $x=0(.1)5$ , 4–9D; ${\mathsf{i}^{(1)}_{n}}\left(x\right)$ , $\mathsf{k}_{n}\left(x\right)$ , $n=0(1)20(10)50$ , 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 $\mathsf{j}_{n}\left(x\right)$ , $\mathsf{j}_{n}'\left(x\right)$ , $\mathsf{y}_{n}\left(x\right)$ , $\mathsf{y}_{n}'\left(x\right)$ , ${\mathsf{i}^{(1)}_{n}}\left(x\right)$ , ${\mathsf{i}^{(1)}_{n}}'\left(x\right)$ , $\mathsf{k}_{n}\left(x\right)$ , $\mathsf{k}_{n}'\left(x\right)$ , $n=0(1)10(10)30$ , 50, 100, $x=1$ , 5, 10, 25, 50, 100, 8S; $x\mathsf{j}_{n}\left(x\right)$ , $(x\mathsf{j}_{n}\left(x\right))^{\prime}$ , $x\mathsf{y}_{n}\left(x\right)$ , $(x\mathsf{y}_{n}\left(x\right))^{\prime}$ (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 $\mathsf{j}_{n}\left(z\right)$ , $\mathsf{j}_{n}'\left(z\right)$ , $\mathsf{y}_{n}\left(z\right)$ , $\mathsf{y}_{n}'\left(z\right)$ , ${\mathsf{i}^{(1)}_{n}}\left(z\right)$ , ${\mathsf{i}^{(1)}_{n}}'\left(z\right)$ , $\mathsf{k}_{n}\left(z\right)$ , $\mathsf{k}_{n}'\left(z\right)$ , $n=0(1)15$ , 20(10)50, 100, $z=4+2i$ , $20+10i$ , 8S. (For the notation replace $j, y, i, k$ by $\mathsf{j}$ , $\mathsf{y}$ , ${\mathsf{i}^{(1)}}$ , $\mathsf{k}$ , 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 $a^{\prime}_{n,m}$ , $\mathsf{j}_{n}\left(a^{\prime}_{n,m}\right)$ , $b^{\prime}_{n,m}$ , $\mathsf{y}_{n}\left(b^{\prime}_{n,m}\right)$ , $n=1(1)20$ , $m=1(1)50$ , 8D. Also included are tables of the coefficients in the uniform asymptotic expansions of these zeros and associated values as $n\to\infty$ .

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