spherical polar coordinates

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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, $L^2$ is the squared angular momentum operator: … ►

2: 1.5 Calculus of Two or More Variables

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§1.5(ii) Coordinate Systems

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Polar Coordinates

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Cylindrical Coordinates

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Spherical Coordinates

… ►For applications and other coordinate systems see §§12.17, 14.19(i), 14.30(iv), 28.32, 29.18, 30.13, 30.14. …

3: 29.18 Mathematical Applications

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§29.18(i) Sphero-Conal Coordinates

… ►(29.18.5) is the differential equation of spherical Bessel functions (§10.47(i)), and (29.18.6), (29.18.7) agree with the Lamé equation (29.2.1). ►

§29.18(ii) Ellipsoidal Coordinates

►The wave equation (29.18.1), when transformed to ellipsoidal coordinates $\alpha ,\beta ,\gamma$: … ►

§29.18(iii) Spherical and Ellipsoidal Harmonics

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4: 10.73 Physical Applications

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§10.73(ii) Spherical Bessel Functions

►The functions ${\mathsf{j}}_n\left(x\right)$, ${\mathsf{y}}_n\left(x\right)$, ${\mathsf{h}}_n^{(1)}\left(x\right)$, and ${\mathsf{h}}_n^{(2)}\left(x\right)$ arise in the solution (again by separation of variables) of the Helmholtz equation in spherical coordinates $\rho ,\theta ,\varphi$ (§1.5(ii)): …With the spherical harmonic $Y_{\mathrm{\ell },m}\left(\theta ,\varphi \right)$ defined as in §14.30(i), the solutions are of the form $f=g_{\mathrm{\ell }}(k\rho )Y_{\mathrm{\ell },m}\left(\theta ,\varphi \right)$ with $g_{\mathrm{\ell }}={\mathsf{j}}_{\mathrm{\ell }}$, ${\mathsf{y}}_{\mathrm{\ell }}$, ${\mathsf{h}}_{\mathrm{\ell }}^{(1)}$, or ${\mathsf{h}}_{\mathrm{\ell }}^{(2)}$, depending on the boundary conditions. Accordingly, the spherical Bessel functions appear in all problems in three dimensions with spherical symmetry involving the scattering of electromagnetic radiation. …

5: 22.18 Mathematical Applications

… ►In polar coordinates, $x=r\cos \varphi$, $y=r\sin \varphi$, the lemniscate is given by $r^2=\cos \left(2\varphi \right)$, $0\le \varphi \le 2\pi$. … ►Discussion of parametrization of the angles of spherical trigonometry in terms of Jacobian elliptic functions is given in Greenhill (1959, p. 131) and Lawden (1989, §4.4). …

6: 10.55 Continued Fractions

§10.55 Continued Fractions

►For continued fractions for ${\mathsf{j}}_{n+1}\left(z\right)/{\mathsf{j}}_n\left(z\right)$ and ${\mathsf{i}}_{n+1}^{(1)}\left(z\right)/{\mathsf{i}}_n^{(1)}\left(z\right)$ see Cuyt et al. (2008, pp. 350, 353, 362, 363, 367–369).

7: 10.48 Graphs

§10.48 Graphs

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Figure 10.48.5: ${\mathsf{i}}_0^{(1)}\left(x\right)$, ${\mathsf{i}}_0^{(2)}\left(x\right)$, ${\mathsf{k}}_0\left(x\right)$, $0\le x\le 4$. Magnify

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Figure 10.48.6: ${\mathsf{i}}_1^{(1)}\left(x\right),{\mathsf{i}}_1^{(2)}\left(x\right),{\mathsf{k}}_1\left(x\right)$, $0\le x\le 4$. Magnify

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Figure 10.48.7: ${\mathsf{i}}_5^{(1)}\left(x\right)$, ${\mathsf{i}}_5^{(2)}\left(x\right)$, ${\mathsf{k}}_5\left(x\right)$, $0\le x\le 8$. Magnify

8: 14.31 Other Applications

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§14.31(i) Toroidal Functions

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§14.31(ii) Conical Functions

►The conical functions ${\mathsf{P}}_{-\frac{1}{2}+\mathrm{i}\tau }^m\left(x\right)$ appear in boundary-value problems for the Laplace equation in toroidal coordinates (§14.19(i)) for regions bounded by cones, by two intersecting spheres, or by one or two confocal hyperboloids of revolution (Kölbig (1981)). … ►Many additional physical applications of Legendre polynomials and associated Legendre functions include solution of the Helmholtz equation, as well as the Laplace equation, in spherical coordinates (Temme (1996b)), quantum mechanics (Edmonds (1974)), and high-frequency scattering by a sphere (Nussenzveig (1965)). …

9: 10.50 Wronskians and Cross-Products

§10.50 Wronskians and Cross-Products

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$$𝒲\left\{{\mathsf{h}}_n^{(1)}\left(z\right),{\mathsf{h}}_n^{(2)}\left(z\right)\right\}=-2\mathrm{i}{z}^{-2}.$$

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$$𝒲\left\{{\mathsf{i}}_n^{(1)}\left(z\right),{\mathsf{i}}_n^{(2)}\left(z\right)\right\}={(-1)}^{n+1}{z}^{-2},$$

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$$𝒲\left\{{\mathsf{i}}_n^{(1)}\left(z\right),{\mathsf{k}}_n\left(z\right)\right\}=𝒲\left\{{\mathsf{i}}_n^{(2)}\left(z\right),{\mathsf{k}}_n\left(z\right)\right\}=-\frac{1}{2}\pi z^{-2}.$$

… ►Results corresponding to (10.50.3) and (10.50.4) for ${\mathsf{i}}_n^{(1)}\left(z\right)$ and ${\mathsf{i}}_n^{(2)}\left(z\right)$ are obtainable via (10.47.12).

10: 10.47 Definitions and Basic Properties

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Equation (10.47.1)

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Equation (10.47.2)

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§10.47(iv) Interrelations

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