on general sets

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11: 23.1 Special Notation

$\mathbb{L}$	lattice in $\mathbb{C}$ .
…
$2\omega_{1},2\omega_{3}$	lattice generators ( $\Im\left(\omega_{3}/\omega_{1}\right)>0$ ).
…

…

12: 10.13 Other Differential Equations

… ►

10.13.3

w^{\prime\prime}+\lambda^{2}z^{p-2}w=0,

w=z^{\frac{1}{2}}\mathscr{C}_{1/p}\left(2\lambda z^{\frac{1}{2}p}/p\right)

ⓘ

Symbols:: $\mathscr{C}_{\NVar{\nu}}\left(\NVar{z}\right)$ : cylinder function and $z$ : complex variable
A&S Ref:: 9.1.51
Permalink:: http://dlmf.nist.gov/10.13.E3
Encodings:: pMML, png, TeX
See also:: Annotations for §10.13 and Ch.10

►

10.13.4

w^{\prime\prime}+\frac{1\mp 2\nu}{z}w^{\prime}+\lambda^{2}w=0,

w=z^{\pm\nu}\mathscr{C}_{\nu}\left(\lambda z\right)

ⓘ

Symbols:: $\mathscr{C}_{\NVar{\nu}}\left(\NVar{z}\right)$ : cylinder function, $z$ : complex variable and $\nu$ : complex parameter
A&S Ref:: 9.1.52
Referenced by:: Erratum (V1.0.8) for Equation (10.13.4)
Permalink:: http://dlmf.nist.gov/10.13.E4
Encodings:: pMML, png, TeX
Generalization (effective with 1.0.8):: This equation has been generalized to allow an additional case. Originally only the case $w=z^{\nu}\mathscr{C}_{\nu}\left(\lambda z\right)$ was covered.
Suggested 2014-01-27 by Tom Koornwinder
See also:: Annotations for §10.13 and Ch.10

… ►

10.13.8

w^{(2n)}=(-1)^{n}\lambda^{2n}z^{-n}w,

w=z^{\frac{1}{2}n}\mathscr{C}_{n}\left(2\lambda e^{k\pi\mathrm{i}/n}z^{\frac{1% }{2}}\right)

k=0,1,\dotsc,2n-1

ⓘ

Symbols:: $\pi$ : the ratio of the circumference of a circle to its diameter, $\mathscr{C}_{\NVar{\nu}}\left(\NVar{z}\right)$ : cylinder function, $\mathrm{e}$ : base of natural logarithm, $\mathrm{i}$ : imaginary unit, $n$ : integer, $k$ : nonnegative integer and $z$ : complex variable
A&S Ref:: 9.1.56
Referenced by:: §10.36
Permalink:: http://dlmf.nist.gov/10.13.E8
Encodings:: pMML, png, TeX
See also:: Annotations for §10.13 and Ch.10

►In (10.13.9)–(10.13.11)

\mathscr{C}_{\nu}\left(z\right)

\mathscr{D}_{\mu}(z)

are any cylinder functions of orders

\nu,\mu

, respectively, and

\vartheta=z(\!\ifrac{\mathrm{d}}{\mathrm{d}z})

. … ►

10.13.11

\left(\vartheta^{4}-2(\nu^{2}+\mu^{2})\vartheta^{2}+(\nu^{2}-\mu^{2})^{2}% \right)w+4z^{2}(\vartheta+1)(\vartheta+2)w=0,

w=\mathscr{C}_{\nu}\left(z\right)\mathscr{D}_{\mu}(z).

ⓘ

Symbols:: $\mathscr{C}_{\NVar{\nu}}\left(\NVar{z}\right)$ : cylinder function, $z$ : complex variable, $\nu$ : complex parameter, $\vartheta=z(\ifrac{\mathrm{d}}{\mathrm{d}z})$ and $\mathscr{D}_{\nu}(z)$ : cylinder function
A&S Ref:: 9.1.57
Referenced by:: §10.13, §10.36
Permalink:: http://dlmf.nist.gov/10.13.E11
Encodings:: pMML, png, TeX
See also:: Annotations for §10.13 and Ch.10

…

13: 26.8 Set Partitions: Stirling Numbers

§26.8 Set Partitions: Stirling Numbers

… ►uniformly for

n=o\left(k^{1/2}\right)

. … ►For other asymptotic approximations and also expansions see Moser and Wyman (1958a) for Stirling numbers of the first kind, and Moser and Wyman (1958b), Bleick and Wang (1974) for Stirling numbers of the second kind. ►For asymptotic estimates for generalized Stirling numbers see Chelluri et al. (2000).

14: 4.24 Inverse Trigonometric Functions: Further Properties

… ►

4.24.13

\operatorname{Arcsin}u\pm\operatorname{Arcsin}v=\operatorname{Arcsin}\left(u(1% -v^{2})^{1/2}\pm v(1-u^{2})^{1/2}\right),

ⓘ

Symbols:: $\operatorname{Arcsin}\NVar{z}$ : general arcsine function
A&S Ref:: 4.4.32
Permalink:: http://dlmf.nist.gov/4.24.E13
Encodings:: pMML, png, TeX
See also:: Annotations for §4.24(iii), §4.24 and Ch.4

►

4.24.14

\operatorname{Arccos}u\pm\operatorname{Arccos}v=\operatorname{Arccos}\left(uv% \mp((1-u^{2})(1-v^{2}))^{1/2}\right),

ⓘ

Symbols:: $\operatorname{Arccos}\NVar{z}$ : general arccosine function
A&S Ref:: 4.4.33
Permalink:: http://dlmf.nist.gov/4.24.E14
Encodings:: pMML, png, TeX
See also:: Annotations for §4.24(iii), §4.24 and Ch.4

… ►

4.24.16

\operatorname{Arcsin}u\pm\operatorname{Arccos}v=\operatorname{Arcsin}\left(uv% \pm((1-u^{2})(1-v^{2}))^{1/2}\right)=\operatorname{Arccos}\left(v(1-u^{2})^{1/% 2}\mp u(1-v^{2})^{1/2}\right),

ⓘ

Symbols:: $\operatorname{Arccos}\NVar{z}$ : general arccosine function and $\operatorname{Arcsin}\NVar{z}$ : general arcsine function
A&S Ref:: 4.4.35
Permalink:: http://dlmf.nist.gov/4.24.E16
Encodings:: pMML, png, TeX
See also:: Annotations for §4.24(iii), §4.24 and Ch.4

…

15: 1.18 Linear Second Order Differential Operators and Eigenfunction Expansions

… ►A (finite or countably infinite, generalizing the definition of (1.2.40)) set

\{v_{n}\}

is an orthonormal set if the

v_{n}

are normalized and pairwise orthogonal. … ►More generally, for

f\in C(X)

x\in X

, see (1.4.24), … ►More generally, continuous spectra may occur in sets of disjoint finite intervals

[\lambda_{a},\lambda_{b}]\in(0,\infty)

, often called bands, when

q(x)

is periodic, see Ashcroft and Mermin (1976, Ch 8) and Kittel (1996, Ch 7). …

16: 16.8 Differential Equations

… ►When no

b_{j}

is an integer, and no two

b_{j}

differ by an integer, a fundamental set of solutions of (16.8.3) is given by … ►

16.8.8

{{}_{q+1}F_{q}}\left({a_{1},\dots,a_{q+1}\atop b_{1},\dots,b_{q}};z\right)=% \sum_{j=1}^{q+1}\left({\textstyle\ifrac{\prod\limits_{\begin{subarray}{c}k=1\\ k\neq j\end{subarray}}^{q+1}\frac{\Gamma\left(a_{k}-a_{j}\right)}{\Gamma\left(% a_{k}\right)}}{\prod\limits_{k=1}^{q}\frac{\Gamma\left(b_{k}-a_{j}\right)}{% \Gamma\left(b_{k}\right)}}}\right)\widetilde{w}_{j}(z),

|\operatorname{ph}\left(-z\right)|\leq\pi

ⓘ

Symbols:: $\Gamma\left(\NVar{z}\right)$ : gamma function, ${{}_{\NVar{p}}F_{\NVar{q}}}\left(\NVar{a_{1},\dots,a_{p}};\NVar{b_{1},\dots,b_% {q}};\NVar{z}\right)$ or ${{}_{\NVar{p}}F_{\NVar{q}}}\left({\NVar{a_{1},\dots,a_{p}}\atop\NVar{b_{1},% \dots,b_{q}}};\NVar{z}\right)$ : alternatively ${{}_{\NVar{p}}F_{\NVar{q}}}\left(\NVar{\mathbf{a}};\NVar{\mathbf{b}};\NVar{z}\right)$ or ${{}_{\NVar{p}}F_{\NVar{q}}}\left({\NVar{\mathbf{a}}\atop\NVar{\mathbf{b}}};% \NVar{z}\right)$
generalized hypergeometric function, $\pi$ : the ratio of the circumference of a circle to its diameter, $\operatorname{ph}$ : phase, $q$ : nonnegative integer, $z$ : complex variable, $a,a_{1},\ldots,a_{p}$ : real or complex parameters, $b,b_{1},\ldots,b_{q}$ : real or complex parameters and $w$ : solution
Referenced by:: §16.11(ii)
Permalink:: http://dlmf.nist.gov/16.8.E8
Encodings:: pMML, png, TeX
See also:: Annotations for §16.8(ii), §16.8 and Ch.16

… ►

16.8.9

\left({\textstyle\ifrac{\prod\limits_{k=1}^{q+1}\Gamma\left(a_{k}\right)}{% \prod\limits_{k=1}^{q}\Gamma\left(b_{k}\right)}}\right){{}_{q+1}F_{q}}\left({a% _{1},\dots,a_{q+1}\atop b_{1},\dots,b_{q}};z\right)=\sum_{j=1}^{q+1}\left(z_{0% }-z\right)^{-a_{j}}\sum_{n=0}^{\infty}\frac{\Gamma\left(a_{j}+n\right)}{n!}\*% \left({\textstyle\ifrac{\prod\limits_{\begin{subarray}{c}k=1\\ k\neq j\end{subarray}}^{q+1}\Gamma\left(a_{k}-a_{j}-n\right)}{\prod\limits_{k=% 1}^{q}\Gamma\left(b_{k}-a_{j}-n\right)}}\right)\*{{}_{q+1}F_{q}}\left({a_{1}-a% _{j}-n,\dots,a_{q+1}-a_{j}-n\atop b_{1}-a_{j}-n,\dots,b_{q}-a_{j}-n};z_{0}% \right)\left(z-z_{0}\right)^{-n}.

ⓘ

Symbols:: $\Gamma\left(\NVar{z}\right)$ : gamma function, ${{}_{\NVar{p}}F_{\NVar{q}}}\left(\NVar{a_{1},\dots,a_{p}};\NVar{b_{1},\dots,b_% {q}};\NVar{z}\right)$ or ${{}_{\NVar{p}}F_{\NVar{q}}}\left({\NVar{a_{1},\dots,a_{p}}\atop\NVar{b_{1},% \dots,b_{q}}};\NVar{z}\right)$ : alternatively ${{}_{\NVar{p}}F_{\NVar{q}}}\left(\NVar{\mathbf{a}};\NVar{\mathbf{b}};\NVar{z}\right)$ or ${{}_{\NVar{p}}F_{\NVar{q}}}\left({\NVar{\mathbf{a}}\atop\NVar{\mathbf{b}}};% \NVar{z}\right)$
generalized hypergeometric function, $!$ : factorial (as in $n!$ ), $q$ : nonnegative integer, $z$ : complex variable, $a,a_{1},\ldots,a_{p}$ : real or complex parameters and $b,b_{1},\ldots,b_{q}$ : real or complex parameters
Referenced by:: §16.8(ii)
Permalink:: http://dlmf.nist.gov/16.8.E9
Encodings:: pMML, png, TeX
See also:: Annotations for §16.8(ii), §16.8 and Ch.16

…

17: 28.33 Physical Applications

… ►with

W(x,y,t)=e^{\mathrm{i}\omega t}V(x,y)

, reduces to (28.32.2) with

k^{2}=\omega^{2}\rho/{\tau}

. …If we denote the positive solutions

q

of (28.33.3) by

q_{n,m}

, then the vibration of the membrane is given by

\omega^{2}_{n,m}=\ifrac{4q_{n,m}\tau}{(c^{2}\rho)}

. The general solution of the problem is a superposition of the separated solutions. … ►Substituting

z=\omega t

a=\ifrac{b}{\omega^{2}}

, and

2q=\ifrac{f}{\omega^{2}}

, we obtain Mathieu’s standard form (28.2.1). ►As

\omega

runs from

0

+\infty

, with

b

and

f

fixed, the point

(q,a)

moves from

\infty

0

along the ray

\mathcal{L}

given by the part of the line

a=(2b/f)q

that lies in the first quadrant of the

(q,a)

-plane. …

18: 11.11 Asymptotic Expansions of Anger–Weber Functions

… ►If

z

is fixed, and

\nu\to\infty

|\operatorname{ph}\nu|\leq\pi

in such a way that

\nu

is bounded away from the set of all integers, then … ►

§11.11(iii) Large $\nu$ , Fixed $z/\nu$

… ►In general, … ►In general, … ►(Note that Olver’s definition of

\mathbf{A}_{\nu}\left(z\right)

omits the factor

1/\pi

in (11.10.4).) …

19: 4.38 Inverse Hyperbolic Functions: Further Properties

… ►

4.38.15

\operatorname{Arcsinh}u\pm\operatorname{Arcsinh}v=\operatorname{Arcsinh}\left(% u(1+v^{2})^{1/2}\pm v(1+u^{2})^{1/2}\right),

ⓘ

Symbols:: $\operatorname{Arcsinh}\NVar{z}$ : general inverse hyperbolic sine function
A&S Ref:: 4.6.26
Permalink:: http://dlmf.nist.gov/4.38.E15
Encodings:: pMML, png, TeX
See also:: Annotations for §4.38(iii), §4.38 and Ch.4

►

4.38.16

\operatorname{Arccosh}u\pm\operatorname{Arccosh}v=\operatorname{Arccosh}\left(% uv\pm((u^{2}-1)(v^{2}-1))^{1/2}\right),

ⓘ

Symbols:: $\operatorname{Arccosh}\NVar{z}$ : general inverse hyperbolic cosine function
A&S Ref:: 4.6.27
Permalink:: http://dlmf.nist.gov/4.38.E16
Encodings:: pMML, png, TeX
See also:: Annotations for §4.38(iii), §4.38 and Ch.4

… ►

4.38.18

\operatorname{Arcsinh}u\pm\operatorname{Arccosh}v=\operatorname{Arcsinh}\left(% uv\pm((1+u^{2})(v^{2}-1))^{1/2}\right)=\operatorname{Arccosh}\left(v(1+u^{2})^% {1/2}\pm u(v^{2}-1)^{1/2}\right),

ⓘ

Symbols:: $\operatorname{Arccosh}\NVar{z}$ : general inverse hyperbolic cosine function and $\operatorname{Arcsinh}\NVar{z}$ : general inverse hyperbolic sine function
A&S Ref:: 4.6.29
Permalink:: http://dlmf.nist.gov/4.38.E18
Encodings:: pMML, png, TeX
See also:: Annotations for §4.38(iii), §4.38 and Ch.4

…

20: 16.1 Special Notation

… ► ►►►►

$p, q$	nonnegative integers.
…
$D$	$\ifrac{\mathrm{d}}{\mathrm{d}z}$ .
$\vartheta$	$z\ifrac{\mathrm{d}}{\mathrm{d}z}$ .

►The main functions treated in this chapter are the generalized hypergeometric function

{{}_{p}F_{q}}\left({a_{1},\dots,a_{p}\atop b_{1},\dots,b_{q}};z\right)

, the Appell (two-variable hypergeometric) functions

{F_{1}}\left(\alpha;\beta,\beta^{\prime};\gamma;x,y\right)

{F_{2}}\left(\alpha;\beta,\beta^{\prime};\gamma,\gamma^{\prime};x,y\right)

{F_{3}}\left(\alpha,\alpha^{\prime};\beta,\beta^{\prime};\gamma;x,y\right)

{F_{4}}\left(\alpha,\beta;\gamma,\gamma^{\prime};x,y\right)

, and the Meijer

G

-function

{G^{m,n}_{p,q}}\left(z;{a_{1},\dots,a_{p}\atop b_{1},\dots,b_{q}}\right)

. Alternative notations are

{{}_{p}F_{q}}\left({\mathbf{a}\atop\mathbf{b}};z\right)

{{}_{p}F_{q}}\left(a_{1},\dots,a_{p};b_{1},\dots,b_{q};z\right)