# metric coefficients

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##### 1: 30.13 Wave Equation in Prolate Spheroidal Coordinates
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###### §30.13(ii) MetricCoefficients
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30.13.3 $h_{\xi}^{2}=\left(\frac{\partial x}{\partial\xi}\right)^{2}+\left(\frac{% \partial y}{\partial\xi}\right)^{2}+\left(\frac{\partial z}{\partial\xi}\right% )^{2}=\frac{c^{2}(\xi^{2}-\eta^{2})}{\xi^{2}-1},$
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30.13.4 $h_{\eta}^{2}=\left(\frac{\partial x}{\partial\eta}\right)^{2}+\left(\frac{% \partial y}{\partial\eta}\right)^{2}+\left(\frac{\partial z}{\partial\eta}% \right)^{2}=\frac{c^{2}(\xi^{2}-\eta^{2})}{1-\eta^{2}},$
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30.13.5 $h_{\phi}^{2}=\left(\frac{\partial x}{\partial\phi}\right)^{2}+\left(\frac{% \partial y}{\partial\phi}\right)^{2}+\left(\frac{\partial z}{\partial\phi}% \right)^{2}=c^{2}(\xi^{2}-1)(1-\eta^{2}).$
βΊIn most applications the solution $w$ has to be a single-valued function of $(x,y,z)$, which requires $\mu=m$ (a nonnegative integer) and …
##### 2: 30.14 Wave Equation in Oblate Spheroidal Coordinates
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###### §30.14(ii) MetricCoefficients
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30.14.5 $h_{\phi}^{2}=c^{2}(\xi^{2}+1)(1-\eta^{2}).$
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30.14.8 $w_{1}(\xi)=a_{1}S^{m(1)}_{n}\left(i\xi,\gamma\right)+b_{1}S^{m(2)}_{n}\left(i% \xi,\gamma\right).$
##### 3: Bibliography K
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• T. A. Kaeding (1995) Pascal program for generating tables of $\mathrm{SU}(3)$ Clebsch-Gordan coefficients. Comput. Phys. Comm. 85 (1), pp. 82–88.
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• P. L. Kapitsa (1951a) Heat conduction and diffusion in a fluid medium with a periodic flow. I. Determination of the wave transfer coefficient in a tube, slot, and canal. Akad. Nauk SSSR. Ε½urnal Eksper. Teoret. Fiz. 21, pp. 964–978.
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• R. P. Kerr (1963) Gravitational field of a spinning mass as an example of algebraically special metrics. Phys. Rev. Lett. 11 (5), pp. 237–238.
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• T. H. Koornwinder (1981) Clebsch-Gordan coefficients for ${\rm SU}(2)$ and Hahn polynomials. Nieuw Arch. Wisk. (3) 29 (2), pp. 140–155.
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##### 5: 26.21 Tables
###### §26.21 Tables
βΊAbramowitz and Stegun (1964, Chapter 24) tabulates binomial coefficients $\genfrac{(}{)}{0.0pt}{}{m}{n}$ for $m$ up to 50 and $n$ up to 25; extends Table 26.4.1 to $n=10$; tabulates Stirling numbers of the first and second kinds, $s\left(n,k\right)$ and $S\left(n,k\right)$, for $n$ up to 25 and $k$ up to $n$; tabulates partitions $p\left(n\right)$ and partitions into distinct parts $p\left(\mathcal{D},n\right)$ for $n$ up to 500. … βΊGoldberg et al. (1976) contains tables of binomial coefficients to $n=100$ and Stirling numbers to $n=40$.
##### 6: 28.14 Fourier Series
βΊThe coefficients satisfy βΊ βΊ
28.14.5 $\sum_{m=-\infty}^{\infty}\left(c_{2m}^{\nu}(q)\right)^{2}=1;$
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##### 7: 26.4 Lattice Paths: Multinomial Coefficients and Set Partitions
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###### §26.4(i) Definitions
βΊ $M_{1}$ is the multinominal coefficient (26.4.2): … βΊ
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##### 8: 29.20 Methods of Computation
βΊSubsequently, formulas typified by (29.6.4) can be applied to compute the coefficients of the Fourier expansions of the corresponding Lamé functions by backward recursion followed by application of formulas typified by (29.6.5) and (29.6.6) to achieve normalization; compare §3.6. … βΊA third method is to approximate eigenvalues and Fourier coefficients of Lamé functions by eigenvalues and eigenvectors of finite matrices using the methods of §§3.2(vi) and 3.8(iv). …The approximations converge geometrically (§3.8(i)) to the eigenvalues and coefficients of Lamé functions as $n\to\infty$. … βΊ
###### §29.20(ii) Lamé Polynomials
βΊThe corresponding eigenvectors yield the coefficients in the finite Fourier series for Lamé polynomials. …
##### 9: 15.7 Continued Fractions
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15.7.1 $\frac{\mathbf{F}\left(a,b;c;z\right)}{\mathbf{F}\left(a,b+1;c+1;z\right)}=t_{0% }-\cfrac{u_{1}z}{t_{1}-\cfrac{u_{2}z}{t_{2}-\cfrac{u_{3}z}{t_{3}-\cdots}}},$
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15.7.3 $\frac{\mathbf{F}\left(a,b;c;z\right)}{\mathbf{F}\left(a,b+1;c+1;z\right)}=v_{0% }-\cfrac{w_{1}}{v_{1}-\cfrac{w_{2}}{v_{2}-\cfrac{w_{3}}{v_{3}-\cdots}}},$
βΊwhere …
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