functions s(ϵ,ℓ;r),c(ϵ,ℓ;r)

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21: 28 Mathieu Functions and Hill’s Equation

Chapter 28 Mathieu Functions and Hill’s Equation

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22: 22.21 Tables

§22.21 Tables

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23: 9.1 Special Notation

… ►Other notations that have been used are as follows: $\mathrm{Ai}\left(-x\right)$ and $\mathrm{Bi}\left(-x\right)$ for $\mathrm{Ai}\left(x\right)$ and $\mathrm{Bi}\left(x\right)$ (Jeffreys (1928), later changed to $\mathrm{Ai}\left(x\right)$ and $\mathrm{Bi}\left(x\right)$); $U(x)=\sqrt{\pi }\mathrm{Bi}\left(x\right)$, $V(x)=\sqrt{\pi }\mathrm{Ai}\left(x\right)$ (Fock (1945)); $A(x)=3^{-1/3}\pi \mathrm{Ai}\left(-3^{-1/3}x\right)$ (Szegő (1967, §1.81)); $e_0(x)=\pi \mathrm{Hi}(-x)$, ${\tilde{e}}_0(x)=-\pi \mathrm{Gi}(-x)$ (Tumarkin (1959)).

24: 14.31 Other Applications

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§14.31(i) Toroidal Functions

►Applications of toroidal functions include expansion of vacuum magnetic fields in stellarators and tokamaks (van Milligen and López Fraguas (1994)), analytic solutions of Poisson’s equation in channel-like geometries (Hoyles et al. (1998)), and Dirichlet problems with toroidal symmetry (Gil et al. (2000)). … ►

§14.31(ii) Conical Functions

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25: 29.19 Physical Applications

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§29.19(i) Lamé Functions

►Simply-periodic Lamé functions ($\nu$ noninteger) can be used to solve boundary-value problems for Laplace’s equation in elliptical cones. …

26: 23.1 Special Notation

… ►The main functions treated in this chapter are the Weierstrass $\mathrm{\wp }$-function $\mathrm{\wp }\left(z\right)=\mathrm{\wp }\left(z|𝕃\right)=\mathrm{\wp }(z;g_2,g_3)$; the Weierstrass zeta function $\zeta \left(z\right)=\zeta \left(z|𝕃\right)=\zeta (z;g_2,g_3)$; the Weierstrass sigma function $\sigma \left(z\right)=\sigma \left(z|𝕃\right)=\sigma (z;g_2,g_3)$; the elliptic modular function $\lambda \left(\tau \right)$; Klein’s complete invariant $J\left(\tau \right)$; Dedekind’s eta function $\eta \left(\tau \right)$. …

27: 27.13 Functions

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27.13.2 $$g\left(k\right)\ge 2^k+\lfloor \frac{3^k}{2^k}\rfloor -2,$$

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Symbols:

$g\left(k\right)$: Waring’s function, $\lfloor x\rfloor$: floor of $x$ and $k$: positive integer

Referenced by:

§27.13(iii)

Permalink:

http://dlmf.nist.gov/27.13.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §27.13(iii), §27.13 and Ch.27

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27.13.5 $${(\vartheta \left(x\right))}^2=1+\sum _{n=1}^{\mathrm{\infty }}{r}_2\left(n\right)x^n.$$

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Symbols:

$\vartheta \left(x\right)={\theta }_3(0,x)$: alternative notation, $r_k\left(n\right)$: number of squares, $n$: positive integer and $x$: real number

Referenced by:

§27.13(iv)

Permalink:

http://dlmf.nist.gov/27.13.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §27.13(iv), §27.13 and Ch.27

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28: 32.14 Combinatorics

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32.14.1 $$\underset{N\to \mathrm{\infty }}{lim}\mathrm{Prob}\left(\frac{{\mathrm{\ell }}_N(𝝅)-2\sqrt{N}}{N^{1/6}}\le s\right)=F(s),$$

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Symbols:

$𝝅(m)$: permutations, ${\mathrm{\ell }}_N(𝝅)$: length and $F(s)$: distribution function

Permalink:

http://dlmf.nist.gov/32.14.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §32.14 and Ch.32

►where the distribution function $F(s)$ is defined here by ►

32.14.2 $$F(s)=\exp \left(-{\int }_s^{\mathrm{\infty }}(x-s)w^2(x)dx\right),$$

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Symbols:

$dx$: differential of $x$ , $\exp z$: exponential function, $\int$: integral, $x$: real and $F(s)$: distribution function

Referenced by:

§32.14

Permalink:

http://dlmf.nist.gov/32.14.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §32.14 and Ch.32

… ►The distribution function $F(s)$ given by (32.14.2) arises in random matrix theory where it gives the limiting distribution for the normalized largest eigenvalue in the Gaussian Unitary Ensemble of $n\times n$ Hermitian matrices; see Tracy and Widom (1994). …

29: 15.1 Special Notation

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15.1.2 $$\frac{F(a,b;c;z)}{\mathrm{\Gamma }\left(c\right)}=𝐅(a,b;c;z)=𝐅(\genfrac{}{}{0pt}{}{a,b}{c};z)={}_2{}^{}𝐅_1^{}(a,b;c;z),$$

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $F(a,b;c;z)$ or $F(\genfrac{}{}{0pt}{}{a,b}{c};z)$: $={}_2{}^{}F_1^{}(a,b;c;z)$ Gauss’ hypergeometric function, ${}_p{}^{}𝐅_q^{}(𝐚;𝐛;z)$ or ${}_p{}^{}𝐅_q^{}(\genfrac{}{}{0pt}{}{𝐚}{𝐛};z)$: scaled (or Olver’s) generalized hypergeometric function, $𝐅(a,b;c;z)$ or $𝐅(\genfrac{}{}{0pt}{}{a,b}{c};z)$: $={}_2{}^{}𝐅_1^{}(a,b;c;z)$ Olver’s hypergeometric function, $z$: complex variable, $a$: real or complex parameter, $b$: real or complex parameter and $c$: real or complex parameter

Permalink:

http://dlmf.nist.gov/15.1.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §15.1 and Ch.15

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30: 27.14 Unrestricted Partitions

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§27.14(v) Divisibility Properties

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§27.14(vi) Ramanujan’s Tau Function

… ►The 24th power of $\eta \left(\tau \right)$ in (27.14.12) with ${\mathrm{e}}^{2\pi \mathrm{i}\tau }=x$ is an infinite product that generates a power series in $x$ with integer coefficients called Ramanujan’s tau function $\tau \left(n\right)$: ►

27.14.18 $$x\prod _{n=1}^{\mathrm{\infty }}{(1-x^n)}^{24}=\sum _{n=1}^{\mathrm{\infty }}\tau \left(n\right)x^n,$$ .

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Defines:

$\tau \left(n\right)$: Ramanujan’s tau function

Symbols:

$n$: positive integer and $x$: real number

Referenced by:

§27.20, §27.20

Permalink:

http://dlmf.nist.gov/27.14.E18

Encodings:

pMML, png, TeX

See also:

Annotations for §27.14(vi), §27.14 and Ch.27

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