布莱辛-里曼护理学院文凭办理【somewhat微aptao168】2MHDhB

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1: Software Index

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	Open Source													With Book						Commercial
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14.34(iv) ${𝖯}_{-\frac{1}{2}+\mathrm{i}\tau }\left(x\right)$, ${𝖰}_{-\frac{1}{2}+\mathrm{i}\tau }\left(x\right)$, ${\widehat{𝖰}}_{-\frac{1}{2}+\mathrm{i}\tau }^{\mu }\left(x\right)$, $P_{-\frac{1}{2}+\mathrm{i}\tau }\left(x\right)$, $Q_{-\frac{1}{2}+\mathrm{i}\tau }\left(x\right)$					✓									✓			✓
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15.20(ii) ${}_2{}^{}F_1^{}(a,b;c;x)$, $x,a,b,c\in ℝ$	✓		✓		✓	✓						a	✓	✓			✓		✓		✓	✓	a
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25.21(v) ${\mathrm{Li}}_2\left(z\right)$, ${\mathrm{Li}}_s\left(z\right)$	✓		✓		✓	✓	✓		✓			✓	✓	✓			✓				✓	✓	✓
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30.18(ii) Eigenvalues ${\lambda }_n^m\left({\gamma }^2\right)$														✓			✓		✓			✓
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These are collections of software (e.g. libraries) or interactive systems of a somewhat broad scope. Contents may be adapted from research software or may be contributed by project participants who donate their services to the project. The software is made freely available to the public, typically in source code form. While formal support of the collection may not be provided by its developers, within active projects there is often a core group who donate time to consider bug reports and make updates to the collection.

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2: 4.17 Special Values and Limits

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Table 4.17.1: Trigonometric functions: values at multiples of $\frac{1}{12}\pi$.

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$\theta$	$\sin \theta$	$\cos \theta$	$\tan \theta$	$\csc \theta$	$\sec \theta$	$\cot \theta$
…
$\pi /12$	$\frac{1}{4}\sqrt{2}(\sqrt{3}-1)$	$\frac{1}{4}\sqrt{2}(\sqrt{3}+1)$	$2-\sqrt{3}$	$\sqrt{2}(\sqrt{3}+1)$	$\sqrt{2}(\sqrt{3}-1)$	$2+\sqrt{3}$
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$\pi /4$	$\frac{1}{2}\sqrt{2}$	$\frac{1}{2}\sqrt{2}$	$1$	$\sqrt{2}$	$\sqrt{2}$	$1$
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$2\pi /3$	$\frac{1}{2}\sqrt{3}$	$-\frac{1}{2}$	$-\sqrt{3}$	$\frac{2}{3}\sqrt{3}$	$-2$	$-\frac{1}{3}\sqrt{3}$
$3\pi /4$	$\frac{1}{2}\sqrt{2}$	$-\frac{1}{2}\sqrt{2}$	$-1$	$\sqrt{2}$	$-\sqrt{2}$	$-1$
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4.17.3 $$\underset{z\to 0}{lim}\frac{1-\cos z}{z^2}=\frac{1}{2}.$$

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Symbols:

$\cos z$: cosine function and $z$: complex variable

Permalink:

http://dlmf.nist.gov/4.17.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §4.17 and Ch.4

3: 34.5 Basic Properties: $\mathit{6}j$ Symbol

… ►If any lower argument in a $\mathit{6}j$ symbol is $0$, $\frac{1}{2}$, or $1$, then the $\mathit{6}j$ symbol has a simple algebraic form. … ►

34.5.5

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Symbols:

: $\mathit{6}j$ symbol, $j,j_r$: non-negative integers or non-negative integers plus one half. and $J$: sum

Permalink:

http://dlmf.nist.gov/34.5.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §34.5(i), §34.5 and Ch.34

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34.5.6

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Symbols:

: $\mathit{6}j$ symbol, $j,j_r$: non-negative integers or non-negative integers plus one half. and $J$: sum

Permalink:

http://dlmf.nist.gov/34.5.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §34.5(i), §34.5 and Ch.34

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34.5.7

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Symbols:

: $\mathit{6}j$ symbol, $j,j_r$: non-negative integers or non-negative integers plus one half. and $J$: sum

Permalink:

http://dlmf.nist.gov/34.5.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §34.5(i), §34.5 and Ch.34

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34.5.13 $$E(j)={\left((j^2-{(j_2-j_3)}^2)({(j_2+j_3+1)}^2-j^2)(j^2-{(l_2-l_3)}^2)({(l_2+l_3+1)}^2-j^2)\right)}^{\frac{1}{2}}.$$

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Defines:

$E(j)$ (locally)

Symbols:

$j,j_r$: non-negative integers or non-negative integers plus one half. and $l,l_r$: non-negative integers or non-negative integers plus one half.

Permalink:

http://dlmf.nist.gov/34.5.E13

Encodings:

pMML, png, TeX

See also:

Annotations for §34.5(iii), §34.5 and Ch.34

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4: 29.1 Special Notation

… ►The main functions treated in this chapter are the eigenvalues $a_{\nu }^{2m}\left(k^2\right)$, $a_{\nu }^{2m+1}\left(k^2\right)$, $b_{\nu }^{2m+1}\left(k^2\right)$, $b_{\nu }^{2m+2}\left(k^2\right)$, the Lamé functions ${\mathrm{𝐸𝑐}}_{\nu }^{2m}(z,k^2)$, ${\mathrm{𝐸𝑐}}_{\nu }^{2m+1}(z,k^2)$, ${\mathrm{𝐸𝑠}}_{\nu }^{2m+1}(z,k^2)$, ${\mathrm{𝐸𝑠}}_{\nu }^{2m+2}(z,k^2)$, and the Lamé polynomials ${\mathrm{𝑢𝐸}}_{2n}^m(z,k^2)$, ${\mathrm{𝑠𝐸}}_{2n+1}^m(z,k^2)$, ${\mathrm{𝑐𝐸}}_{2n+1}^m(z,k^2)$, ${\mathrm{𝑑𝐸}}_{2n+1}^m(z,k^2)$, ${\mathrm{𝑠𝑐𝐸}}_{2n+2}^m(z,k^2)$, ${\mathrm{𝑠𝑑𝐸}}_{2n+2}^m(z,k^2)$, ${\mathrm{𝑐𝑑𝐸}}_{2n+2}^m(z,k^2)$, ${\mathrm{𝑠𝑐𝑑𝐸}}_{2n+3}^m(z,k^2)$. … ►Other notations that have been used are as follows: Ince (1940a) interchanges $a_{\nu }^{2m+1}\left(k^2\right)$ with $b_{\nu }^{2m+1}\left(k^2\right)$. The relation to the Lamé functions $L_{c\nu }^{(m)}$, $L_{s\nu }^{(m)}$of Jansen (1977) is given by … ►

$${\mathrm{𝐸𝑠}}_{\nu }^{2m+2}(z,k^2)=s_{\nu }^{2m+2}(k^2){\mathrm{Es}}_{\nu }^{2m+2}(z,k^2),$$

►where the positive factors $c_{\nu }^m(k^2)$ and $s_{\nu }^m(k^2)$ are determined by …

5: 4.30 Elementary Properties

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Table 4.30.1: Hyperbolic functions: interrelations. …

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	$\sinh \theta =a$	$\cosh \theta =a$	$\tanh \theta =a$	$\mathrm{csch}\theta =a$	$\mathrm{sech}\theta =a$	$\coth \theta =a$
$\sinh \theta$	$a$	${(a^2-1)}^{1/2}$	$a{(1-a^2)}^{-1/2}$	$a^{-1}$	$a^{-1}{(1-a^2)}^{1/2}$	${(a^2-1)}^{-1/2}$
$\cosh \theta$	${(1+a^2)}^{1/2}$	$a$	${(1-a^2)}^{-1/2}$	$a^{-1}{(1+a^2)}^{1/2}$	$a^{-1}$	$a{(a^2-1)}^{-1/2}$
$\tanh \theta$	$a{(1+a^2)}^{-1/2}$	$a^{-1}{(a^2-1)}^{1/2}$	$a$	${(1+a^2)}^{-1/2}$	${(1-a^2)}^{1/2}$	$a^{-1}$
$\mathrm{csch}\theta$	$a^{-1}$	${(a^2-1)}^{-1/2}$	$a^{-1}{(1-a^2)}^{1/2}$	$a$	$a{(1-a^2)}^{-1/2}$	${(a^2-1)}^{1/2}$
$\mathrm{sech}\theta$	${(1+a^2)}^{-1/2}$	$a^{-1}$	${(1-a^2)}^{1/2}$	$a{(1+a^2)}^{-1/2}$	$a$	$a^{-1}{(a^2-1)}^{1/2}$
…

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6: 29.15 Fourier Series and Chebyshev Series

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Polynomial ${\mathrm{𝑢𝐸}}_{2n}^m(z,k^2)$

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Polynomial ${\mathrm{𝑠𝐸}}_{2n+1}^m(z,k^2)$

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Polynomial ${\mathrm{𝑠𝑐𝐸}}_{2n+2}^m(z,k^2)$

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Polynomial ${\mathrm{𝑠𝑑𝐸}}_{2n+2}^m(z,k^2)$

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Polynomial ${\mathrm{𝑐𝑑𝐸}}_{2n+2}^m(z,k^2)$

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7: 22.9 Cyclic Identities

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§22.9(ii) Typical Identities of Rank 2

… ►These identities are cyclic in the sense that each of the indices $m,n$ in the first product of, for example, the form $s_{m,p}^{(4)}{s}_{n,p}^{(4)}$ are simultaneously permuted in the cyclic order: $m\to m+1\to m+2\to \mathrm{\cdots }p\to 1\to 2\to \mathrm{\cdots }m-1$; $n\to n+1\to n+2\to \mathrm{\cdots }p\to 1\to 2\to \mathrm{\cdots }n-1$. … ►

22.9.11 $${\left(d_{1,2}^{(2)}\right)}^2{d}_{2,2}^{(2)}\pm {\left(d_{2,2}^{(2)}\right)}^2{d}_{1,2}^{(2)}=k^{\prime }\left(d_{1,2}^{(2)}\pm d_{2,2}^{(2)}\right),$$

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Symbols:

$k^{\prime }$: complementary modulus and $d_{m,p}^{(r)}$: cyclic quantity

Permalink:

http://dlmf.nist.gov/22.9.E11

Encodings:

pMML, png, TeX

See also:

Annotations for §22.9(iii), §22.9(iii), §22.9 and Ch.22

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22.9.12 $$c_{1,2}^{(2)}{s}_{1,2}^{(2)}{d}_{2,2}^{(2)}+c_{2,2}^{(2)}{s}_{2,2}^{(2)}{d}_{1,2}^{(2)}=0.$$

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Symbols:

$s_{m,p}^{(r)}$: cyclic quantity, $c_{m,p}^{(r)}$: cyclic quantity and $d_{m,p}^{(r)}$: cyclic quantity

Permalink:

http://dlmf.nist.gov/22.9.E12

Encodings:

pMML, png, TeX

See also:

Annotations for §22.9(iii), §22.9(iii), §22.9 and Ch.22

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22.9.21 $$k^2{c}_{1,2}^{(2)}{s}_{1,2}^{(2)}{c}_{2,2}^{(2)}{s}_{2,2}^{(2)}=k^{\prime }\left(1-{\left(s_{1,2}^{(2)}\right)}^2-{\left(s_{2,2}^{(2)}\right)}^2\right).$$

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Symbols:

$k$: modulus, $k^{\prime }$: complementary modulus, $s_{m,p}^{(r)}$: cyclic quantity and $c_{m,p}^{(r)}$: cyclic quantity

Permalink:

http://dlmf.nist.gov/22.9.E21

Encodings:

pMML, png, TeX

See also:

Annotations for §22.9(iv), §22.9(iv), §22.9 and Ch.22

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8: 30.3 Eigenvalues

… ►With $\mu =m=0,1,2,\mathrm{\dots }$, the spheroidal wave functions ${\mathrm{𝖯𝗌}}_n^m(x,{\gamma }^2)$ are solutions of Equation (30.2.1) which are bounded on $(-1,1)$, or equivalently, which are of the form ${(1-x^2)}^{\frac{1}{2}m}g(x)$ where $g(z)$ is an entire function of $z$. These solutions exist only for eigenvalues ${\lambda }_n^m\left({\gamma }^2\right)$, $n=m,m+1,m+2,\mathrm{\dots }$, of the parameter $\lambda$. … ►The eigenvalues ${\lambda }_n^m\left({\gamma }^2\right)$ are analytic functions of the real variable ${\gamma }^2$ and satisfy … ►has the solutions $\lambda ={\lambda }_{m+2j}^m\left({\gamma }^2\right)$, $j=0,1,2,\mathrm{\dots }$. If $p$ is an odd positive integer, then Equation (30.3.5) has the solutions $\lambda ={\lambda }_{m+2j+1}^m\left({\gamma }^2\right)$, $j=0,1,2,\mathrm{\dots }$. …

9: 29.6 Fourier Series

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§29.6(i) Function ${\mathrm{𝐸𝑐}}_{\nu }^{2m}(z,k^2)$

… ►In the special case $\nu =2n$, $m=0,1,\mathrm{\dots },n$, there is a unique nontrivial solution with the property $A_{2p}=0$, $p=n+1,n+2,\mathrm{\dots }$. … ►

§29.6(ii) Function ${\mathrm{𝐸𝑐}}_{\nu }^{2m+1}(z,k^2)$

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§29.6(iii) Function ${\mathrm{𝐸𝑠}}_{\nu }^{2m+1}(z,k^2)$

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§29.6(iv) Function ${\mathrm{𝐸𝑠}}_{\nu }^{2m+2}(z,k^2)$

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10: 29.12 Definitions

… ►The Lamé functions ${\mathrm{𝐸𝑐}}_{\nu }^m(z,k^2)$, $m=0,1,\mathrm{\dots },\nu$, and ${\mathrm{𝐸𝑠}}_{\nu }^m(z,k^2)$, $m=1,2,\mathrm{\dots },\nu$, are called the Lamé polynomials. …where $n=0,1,2,\mathrm{\dots }$, $m=0,1,2,\mathrm{\dots },n$. … ►where $\rho$, $\sigma$, $\tau$ are either $0$ or $\frac{1}{2}$. The polynomial $P(\xi )$ is of degree $n$ and has $m$ zeros (all simple) in $(0,1)$ and $n-m$ zeros (all simple) in $(1,k^{-2})$. … ►defined for $(t_1,t_2,\mathrm{\dots },t_n)$ with …