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11: 31.10 Integral Equations and Representations

… ►A further change of variables, to spherical coordinates, … ►

31.10.21 $$\frac{{\partial }^2𝒦}{{\partial r}^2}+\frac{2(\gamma +\delta +ϵ)-1}{r}\frac{\partial 𝒦}{\partial r}+\frac{1}{r^2}\frac{{\partial }^2𝒦}{{\partial \theta }^2}+\frac{(2(\delta +ϵ)-1)\cot \theta -(2\gamma -1)\tan \theta }{r^2}\frac{\partial 𝒦}{\partial \theta }+\frac{1}{r^2{\sin }^2\theta }\frac{{\partial }^2𝒦}{{\partial \varphi }^2}+\frac{(2\delta -1)\cot \varphi -(2ϵ-1)\tan \varphi }{r^2{\sin }^2\theta }\frac{\partial 𝒦}{\partial \varphi }=0.$$

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Symbols:

$\cot z$: cotangent function, $\frac{\partial f}{\partial x}$: partial derivative of $f$ with respect to $x$ , $\partial x$: partial differential of $x$ , $\sin z$: sine function, $\tan z$: tangent function, $\gamma$: real or complex parameter, $\delta$: real or complex parameter, $ϵ$: real or complex parameter, $𝒦(z;s,t)$: kernel, $r$: radius, $\theta$: angle and $\varphi$: angle

Permalink:

http://dlmf.nist.gov/31.10.E21

Encodings:

pMML, png, TeX

See also:

Annotations for §31.10(ii), §31.10(ii), §31.10 and Ch.31

… ►

31.10.22

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Symbols:

: Riemann’s $P$-symbol for solutions of the generalized hypergeometric differential equation, $\cos z$: cosine function, $\sin z$: sine function, $\gamma$: real or complex parameter, $\delta$: real or complex parameter, $ϵ$: real or complex parameter, $m$: nonnegative integer, $a$: complex parameter, $𝒦(z;s,t)$: kernel, $r$: radius, $\theta$: angle, $\varphi$: angle, $b$ and $c$

Permalink:

http://dlmf.nist.gov/31.10.E22

Encodings:

pMML, png, TeX

See also:

Annotations for §31.10(ii), §31.10(ii), §31.10 and Ch.31

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12: 1.10 Functions of a Complex Variable

… ►Let $f(z)$ be analytic on the disk . …The right-hand side is the Taylor series for $f(z)$ at $z=z_0$, and its radius of convergence is at least $R$. … ►In , if $f(z)$ is analytic, $\left|f(z)\right|\le M$, and $f(0)=0$, then ►

1.10.11 $$\left|f(z)\right|\le \frac{M\left|z\right|}{R}\text{ and }\left|f^{\prime }(0)\right|\le \frac{M}{R}.$$

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Symbols:

$z$: variable, $R$: radius and $\left|x\right|$: absolute value of $x$

Permalink:

http://dlmf.nist.gov/1.10.E11

Encodings:

pMML, png, TeX

See also:

Annotations for §1.10(v), §1.10(v), §1.10 and Ch.1

… ►The radius of convergence $R$ might depend on $x$. …

13: 1.11 Zeros of Polynomials

… ►

1.11.23 $$\sqrt[n]{R}\left(\cos \left(\frac{\alpha +2k\pi }{n}\right)+\mathrm{i}\sin \left(\frac{\alpha +2k\pi }{n}\right)\right),$$

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $\mathrm{i}$: imaginary unit, $\sin z$: sine function, $k$: integer, $n$: nonnegative integer and $R$: radius

Permalink:

http://dlmf.nist.gov/1.11.E23

Encodings:

pMML, png, TeX

See also:

Annotations for §1.11(iv), §1.11 and Ch.1

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14: 16.2 Definition and Analytic Properties

… ►If none of the $a_j$ is a nonpositive integer, then the radius of convergence of the series (16.2.1) is $1$, and outside the open disk the generalized hypergeometric function is defined by analytic continuation with respect to $z$. …

15: 1.6 Vectors and Vector-Valued Functions

… ►For a sphere $x=\rho \sin \theta \cos \varphi$, $y=\rho \sin \theta \sin \varphi$, $z=\rho \cos \theta$, ►

1.6.50 $$\Vert {𝐓}_{\theta }\times {𝐓}_{\varphi }\Vert ={\rho }^2\left|\sin \theta \right|.$$

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Symbols:

$\sin z$: sine function, $\Vert 𝐯\Vert$: vector norm ($l$²), $\rho$: radius, $\theta$: angle, $\varphi$: angle and $\left|x\right|$: absolute value of $x$

Permalink:

http://dlmf.nist.gov/1.6.E50

Encodings:

pMML, png, TeX

See also:

Annotations for §1.6(v), §1.6 and Ch.1

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16: 4.13 Lambert $W$-Function

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