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21: Howard S. Cohl

… ►Cohl has published papers in orthogonal polynomials and special functions, and is particularly interested in fundamental solutions of linear partial differential equations on Riemannian manifolds, associated Legendre functions, generalized and basic hypergeometric functions, eigenfunction expansions of fundamental solutions in separable coordinate systems for linear partial differential equations, orthogonal polynomial generating function and generalized expansions, and $q$-series. …

22: 32.13 Reductions of Partial Differential Equations

§32.13 Reductions of Partial Differential Equations

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23: 4.22 Infinite Products and Partial Fractions

§4.22 Infinite Products and Partial Fractions

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24: 4.36 Infinite Products and Partial Fractions

§4.36 Infinite Products and Partial Fractions

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25: 21.9 Integrable Equations

… ►Here, and in what follows, $x,y$, and $t$ suffixes indicate partial derivatives. … ►

21.9.4 $$u(x,y,t)=c+2\frac{{\partial }^2}{{\partial x}^2}\ln \left(\theta \left(𝐤x+𝐥y+𝝎t+\mathit{\varphi }\right|𝛀)\right),$$

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Symbols:

$\theta \left(𝐳\right|𝛀)$: Riemann theta function, $\ln z$: principal branch of logarithm function, $\frac{\partial f}{\partial x}$: partial derivative of $f$ with respect to $x$ , $\partial x$: partial differential of $x$ , $𝛀$: a Riemann matrix, $u(x,y,t)$: elevation and $c$: complex constant

Referenced by:

§21.9

Permalink:

http://dlmf.nist.gov/21.9.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §21.9 and Ch.21

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26: 7.19 Voigt Functions

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7.19.8 $$𝖵(x,t)=x𝖴(x,t)+2t\frac{\partial 𝖴(x,t)}{\partial x},$$

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Symbols:

$𝖴(x,t)$: Voigt function, $𝖵(x,t)$: Voigt function, $\frac{\partial f}{\partial x}$: partial derivative of $f$ with respect to $x$ , $\partial x$: partial differential of $x$ and $x$: real variable

Referenced by:

§7.19(iii)

Permalink:

http://dlmf.nist.gov/7.19.E8

Encodings:

pMML, png, TeX

See also:

Annotations for §7.19(iii), §7.19 and Ch.7

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7.19.9 $$𝖴(x,t)=1-x𝖵(x,t)-2t\frac{\partial 𝖵(x,t)}{\partial x}.$$

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Symbols:

$𝖴(x,t)$: Voigt function, $𝖵(x,t)$: Voigt function, $\frac{\partial f}{\partial x}$: partial derivative of $f$ with respect to $x$ , $\partial x$: partial differential of $x$ and $x$: real variable

Referenced by:

§7.19(iii)

Permalink:

http://dlmf.nist.gov/7.19.E9

Encodings:

pMML, png, TeX

See also:

Annotations for §7.19(iii), §7.19 and Ch.7

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27: 14.30 Spherical and Spheroidal Harmonics

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14.30.10 $$\frac{1}{{\rho }^2}\frac{\partial }{\partial \rho }\left({\rho }^2\frac{\partial W}{\partial \rho }\right)+\frac{1}{{\rho }^2\sin \theta }\frac{\partial }{\partial \theta }\left(\sin \theta \frac{\partial W}{\partial \theta }\right)+\frac{1}{{\rho }^2{\sin }^2\theta }\frac{{\partial }^{2}W}{{\partial \varphi }^2}=0,$$

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Symbols:

$\frac{\partial f}{\partial x}$: partial derivative of $f$ with respect to $x$ , $\partial x$: partial differential of $x$ , $\sin z$: sine function and $W(\rho ,\theta ,\varphi )$: solution

Permalink:

http://dlmf.nist.gov/14.30.E10

Encodings:

pMML, png, TeX

See also:

Annotations for §14.30(iv), §14.30 and Ch.14

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14.30.12 $$L^2=-{\mathrm{\hslash }}^2\left(\frac{1}{\sin \theta }\frac{\partial }{\partial \theta }\left(\sin \theta \frac{\partial }{\partial \theta }\right)+\frac{1}{{\sin }^2\theta }\frac{{\partial }^2}{{\partial \varphi }^2}\right),$$

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Defines:

$L$: angular momentum operator (locally)

Symbols:

$\frac{\partial f}{\partial x}$: partial derivative of $f$ with respect to $x$ , $\partial x$: partial differential of $x$ and $\sin z$: sine function

Referenced by:

§18.39(ii)

Permalink:

http://dlmf.nist.gov/14.30.E12

Encodings:

pMML, png, TeX

See also:

Annotations for §14.30(iv), §14.30 and Ch.14

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14.30.13 $${\mathrm{L}}_z=-\mathrm{i}\mathrm{\hslash }\frac{\partial }{\partial \varphi };$$

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Defines:

${\mathrm{L}}_z$: $z$ component of the angular momentum operator (locally)

Symbols:

$\mathrm{i}$: imaginary unit, $\frac{\partial f}{\partial x}$: partial derivative of $f$ with respect to $x$ , $\partial x$: partial differential of $x$ and $z$: complex variable

Referenced by:

§14.30(iv), Erratum (V1.1.0) for Additions

Permalink:

http://dlmf.nist.gov/14.30.E13

Encodings:

pMML, png, TeX

Addition (effective with 1.1.0):

This equation was added.

See also:

Annotations for §14.30(iv), §14.30 and Ch.14

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28: 28.33 Physical Applications

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28.33.1 $$\frac{{\partial }^{2}W}{{\partial x}^2}+\frac{{\partial }^{2}W}{{\partial y}^2}-\frac{\rho }{\tau }\frac{{\partial }^{2}W}{{\partial t}^2}=0,$$

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Symbols:

$\frac{\partial f}{\partial x}$: partial derivative of $f$ with respect to $x$ , $\partial x$: partial differential of $x$ , $x$: real variable, $y$: real variable, $\rho$: mass, $\tau$: radial tension and $W(x,y,t)$: solution

Permalink:

http://dlmf.nist.gov/28.33.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §28.33(ii), §28.33 and Ch.28

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29: Ronald F. Boisvert

… ►His research interests include numerical solution of partial differential equations, mathematical software, and information services that support computational science. …

30: Peter A. Clarkson

… ► Kruskal, he developed the “direct method” for determining symmetry solutions of partial differential equations in New similarity reductions of the Boussinesq equation (with M. …