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21: 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 …

22: 1.3 Determinants, Linear Operators, and Spectral Expansions

… ►For $n=2$: … ►for every distinct pair of $j,k$, or when one of the factors ${\sum }_{k=1}^n{a}_{jk}^2$ vanishes. … ►where ${\omega }_1,{\omega }_2,\mathrm{\dots },{\omega }_n$ are the $n$th roots of unity (1.11.21). … ►Let $a_{j,k}$ be defined for all integer values of $j$ and $k$, and ${𝐷}_n[a_{j,k}]$ denote the $(2n+1)\times (2n+1)$ determinant … ►Taking $l^2$ norms, …

23: 27.13 Functions

… ►for all $k\ge 2$, with equality if $4\le k\le 200,000$. If $3^k=q{2}^k+r$ with , then equality holds in (27.13.2) provided $r+q\le 2^k$, a condition that is satisfied with at most a finite number of exceptions. … ►Hardy and Littlewood (1925) conjectures that when $k$ is not a power of 2, and that $G\left(k\right)\le 4k$ when $k$ is a power of 2, but the most that is known (in 2009) is for some constant $c$. … ►Hence $r_2\left(5\right)=8$ because both divisors, $1$ and $5$, are congruent to $1(mod4)$. In fact, there are four representations, given by $5=2^2+1^2=2^2+{(-1)}^2={(-2)}^2+1^2={(-2)}^2+{(-1)}^2$, and four more with the order of summands reversed. …

24: 22.8 Addition Theorems

… ►

22.8.6 $$\mathrm{nd}(u+v)=\frac{\mathrm{nd}u\mathrm{nd}v+k^2\mathrm{sd}u\mathrm{cd}u\mathrm{sd}v\mathrm{cd}v}{1+k^{2}k_{}^{\prime }{}_{}{}^2{\mathrm{sd}}^{2}u{\mathrm{sd}}^{2}v},$$

ⓘ

Symbols:

$\mathrm{cd}(z,k)$: Jacobian elliptic function, $\mathrm{nd}(z,k)$: Jacobian elliptic function, $\mathrm{sd}(z,k)$: Jacobian elliptic function, $u$: complex, $v$: complex and $k$: modulus

Permalink:

http://dlmf.nist.gov/22.8.E6

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See also:

Annotations for §22.8(i), §22.8 and Ch.22

… ►

22.8.21 $$k_{}^{\prime }{}_{}{}^2-k_{}^{\prime }{}_{}{}^2{k}^2\mathrm{sn}{z}_1\mathrm{sn}{z}_2\mathrm{sn}{z}_3\mathrm{sn}{z}_4+k^2\mathrm{cn}{z}_1\mathrm{cn}{z}_2\mathrm{cn}{z}_3\mathrm{cn}{z}_4-\mathrm{dn}{z}_1\mathrm{dn}{z}_2\mathrm{dn}{z}_3\mathrm{dn}{z}_4=0.$$

ⓘ

Symbols:

$\mathrm{cn}(z,k)$: Jacobian elliptic function, $\mathrm{dn}(z,k)$: Jacobian elliptic function, $\mathrm{sn}(z,k)$: Jacobian elliptic function, $z$: complex, $k$: modulus and $k^{\prime }$: complementary modulus

Permalink:

http://dlmf.nist.gov/22.8.E21

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See also:

Annotations for §22.8(iii), §22.8 and Ch.22

… ►

22.8.24 $$z_1-z_2=z_2-z_3=\frac{2}{3}K\left(k\right),$$

ⓘ

Symbols:

$K\left(k\right)$: Legendre’s complete elliptic integral of the first kind, $z$: complex and $k$: modulus

Permalink:

http://dlmf.nist.gov/22.8.E24

Encodings:

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See also:

Annotations for §22.8(iii), §22.8 and Ch.22

… ►is independent of $z_1$, $z_2$, $z_3$. … ►

22.8.26 $$z_1-z_2=z_2-z_3=z_3-z_4=\frac{1}{2}K\left(k\right),$$

ⓘ

Symbols:

$K\left(k\right)$: Legendre’s complete elliptic integral of the first kind, $z$: complex and $k$: modulus

Permalink:

http://dlmf.nist.gov/22.8.E26

Encodings:

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See also:

Annotations for §22.8(iii), §22.8 and Ch.22

…

25: 4.30 Elementary Properties

… ►

Table 4.30.1: Hyperbolic functions: interrelations. …

► ►►►►►►►

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

►

26: 19.30 Lengths of Plane Curves

… ►When $0\le \varphi \le \frac{1}{2}\pi$, … ►

$k^2=1-(b^2/a^2),$

… ►Let $a^2$ and $b^2$ be replaced respectively by $a^2+\lambda$ and $b^2+\lambda$, where $\lambda \in (-b^2,\mathrm{\infty })$, to produce a family of confocal ellipses. … ►See Carlson (1977b, Ex. 9.4-1 and (9.4-4)) for arclengths of hyperbolas and ellipses in terms of $R_{-a}$ that differ only in the sign of $b^2$. … ►For other plane curves with arclength representable by an elliptic integral see Greenhill (1892, p. 190) and Bowman (1953, pp. 32–33). …

27: 26.17 The Twelvefold Way

… ►Table 26.17.1 is reproduced (in modified form) from Stanley (1997, p. 33). … ►

Table 26.17.1: The twelvefold way.

► ►►►

elements of $N$	elements of $K$	$f$ unrestricted	$f$ one-to-one	$f$ onto
…
labeled	unlabeled	$\begin{array}{c}S(n,1)+S(n,2)\\ +\mathrm{\cdots }+S(n,k)\end{array}$		$S(n,k)$
…

►

28: 28.8 Asymptotic Expansions for Large $q$

… ►Denote $h=\sqrt{q}$ and $s=2m+1$. Then as $h\to +\mathrm{\infty }$ with $m=0,1,2,\mathrm{\dots }$, … ►Let $x=\frac{1}{2}\pi +\lambda h^{-1/4}$, where $\lambda$ is a real constant such that . Also let $\xi =2\sqrt{h}\cos x$ and $D_m\left(\xi \right)={\mathrm{e}}^{-{\xi }^2/4}{\mathrm{𝐻𝑒}}_m\left(\xi \right)$ (§18.3). … ►Let $x=\frac{1}{2}\pi -\mu h^{-1/4}$, where $\mu$ is a constant such that $\mu \ge 1$, and $s=2m+1$. …

29: 29.15 Fourier Series and Chebyshev Series

… ►

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

… ►

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

… ►

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

… ►

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

… ►

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

…

30: 22.9 Cyclic Identities

… ►

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

ⓘ

Symbols:

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

Permalink:

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

Encodings:

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See also:

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

►

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

ⓘ

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:

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See also:

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

… ►

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

ⓘ

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

…