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11: 22.10 Maclaurin Series

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22.10.1 $$\mathrm{sn}(z,k)=z-\left(1+k^2\right)\frac{z^3}{3!}+\left(1+14k^2+k^4\right)\frac{z^5}{5!}-\left(1+135k^2+135k^4+k^6\right)\frac{z^7}{7!}+O\left(z^9\right),$$

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Symbols:

$O\left(x\right)$: order not exceeding, $\mathrm{sn}(z,k)$: Jacobian elliptic function, $!$: factorial (as in $n!$), $z$: complex and $k$: modulus

A&S Ref:

16.22.1

Permalink:

http://dlmf.nist.gov/22.10.E1

Encodings:

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See also:

Annotations for §22.10(i), §22.10 and Ch.22

… ►The full expansions converge when . ►

§22.10(ii) Maclaurin Series in $k$ and $k^{\prime }$

… ►The radius of convergence is the distance to the origin from the nearest pole in the complex $k$-plane in the case of (22.10.4)–(22.10.6), or complex $k^{\prime }$-plane in the case of (22.10.7)–(22.10.9); see §22.17.

12: 22.14 Integrals

… ►See §22.16(i) for $\mathrm{am}(z,k)$. … ►Thirdly, with , … ►Lastly, with , … ►In (22.14.13)–(22.14.15), . … ►

22.14.18 $${\int }_0^{K\left(k\right)}\ln \left(\mathrm{dn}(t,k)\right)dt=\frac{1}{2}K\left(k\right)\ln k^{\prime }.$$

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Symbols:

$\mathrm{dn}(z,k)$: Jacobian elliptic function, $K\left(k\right)$: Legendre’s complete elliptic integral of the first kind, $dx$: differential of $x$ , $\int$: integral, $\ln z$: principal branch of logarithm function, $k$: modulus and $k^{\prime }$: complementary modulus

Permalink:

http://dlmf.nist.gov/22.14.E18

Encodings:

pMML, png, TeX

See also:

Annotations for §22.14(iv), §22.14 and Ch.22

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13: 22.3 Graphics

… ►Line graphs of the functions $\mathrm{sn}(x,k)$, $\mathrm{cn}(x,k)$, $\mathrm{dn}(x,k)$, $\mathrm{cd}(x,k)$, $\mathrm{sd}(x,k)$, $\mathrm{nd}(x,k)$, $\mathrm{dc}(x,k)$, $\mathrm{nc}(x,k)$, $\mathrm{sc}(x,k)$, $\mathrm{ns}(x,k)$, $\mathrm{ds}(x,k)$, and $\mathrm{cs}(x,k)$ for representative values of real $x$ and real $k$ illustrating the near trigonometric ($k=0$), and near hyperbolic ($k=1$) limits. … ► $\mathrm{sn}(x,k)$, $\mathrm{cn}(x,k)$, and $\mathrm{dn}(x,k)$ as functions of real arguments $x$ and $k$. The period diverges logarithmically as $k\to 1-$; see §19.12. … ►

§22.3(iii) Complex $z$; Real $k$

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§22.3(iv) Complex $k$

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14: 22.1 Special Notation

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$x,y$	real variables.
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$k$	modulus. Except in §§22.3(iv), 22.17, and 22.19, $0\le k\le 1$.
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$K$, $K^{\prime }$	$K\left(k\right)$, $K^{\prime }\left(k\right)=K\left(k^{\prime }\right)$ (complete elliptic integrals of the first kind (§19.2(ii))).
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… ►The functions treated in this chapter are the three principal Jacobian elliptic functions $\mathrm{sn}(z,k)$, $\mathrm{cn}(z,k)$, $\mathrm{dn}(z,k)$; the nine subsidiary Jacobian elliptic functions $\mathrm{cd}(z,k)$, $\mathrm{sd}(z,k)$, $\mathrm{nd}(z,k)$, $\mathrm{dc}(z,k)$, $\mathrm{nc}(z,k)$, $\mathrm{sc}(z,k)$, $\mathrm{ns}(z,k)$, $\mathrm{ds}(z,k)$, $\mathrm{cs}(z,k)$; the amplitude function $\mathrm{am}(x,k)$; Jacobi’s epsilon and zeta functions $ℰ(x,k)$ and $\mathrm{Z}\left(x|k\right)$. … ►The notation $\mathrm{sn}(z,k)$, $\mathrm{cn}(z,k)$, $\mathrm{dn}(z,k)$ is due to Gudermann (1838), following Jacobi (1827); that for the subsidiary functions is due to Glaisher (1882). Other notations for $\mathrm{sn}(z,k)$ are $\mathrm{sn}(z|m)$ and $\mathrm{sn}(z,m)$ with $m=k^2$; see Abramowitz and Stegun (1964) and Walker (1996). …

15: 26.17 The Twelvefold Way

… ►The twelvefold way gives the number of mappings $f$ from set $N$ of $n$ objects to set $K$ of $k$ objects (putting balls from set $N$ into boxes in set $K$). …In this table ${\left(k\right)}_n$ is Pochhammer’s symbol, and $S(n,k)$ and $p_k\left(n\right)$ are defined in §§26.8(i) and 26.9(i). … ►

Table 26.17.1: The twelvefold way.

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elements of $N$	elements of $K$	$f$ unrestricted	$f$ one-to-one	$f$ onto
labeled	labeled	$k^n$	${\left(k-n+1\right)}_n$	$k!S(n,k)$
unlabeled	labeled	$\left({\displaystyle \genfrac{}{}{0pt}{}{k+n-1}{n}}\right)$	$\left({\displaystyle \genfrac{}{}{0pt}{}{k}{n}}\right)$	$\left({\displaystyle \genfrac{}{}{0pt}{}{n-1}{n-k}}\right)$
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unlabeled	unlabeled	$p_k\left(n\right)$		$p_k\left(n\right)-p_{k-1}\left(n\right)$

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16: 36.6 Scaling Relations

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${\mathrm{\Psi }}_K(𝐱;k)=k^{{\beta }_K}{\mathrm{\Psi }}_K\left(𝐲(k)\right),$

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$\text{cuspoids: }𝐲(k)=(x_1{k}^{{\gamma }_{1K}},x_2{k}^{{\gamma }_{2K}},\mathrm{\dots },x_K{k}^{{\gamma }_{KK}}),$

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Indices for $k$-Scaling of Magnitude of ${\mathrm{\Psi }}_K$ or ${\mathrm{\Psi }}^{(\mathrm{U})}$ (Singularity Index)

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Indices for $k$-Scaling of Coordinates $x_m$

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Indices for $k$-Scaling of $𝐱$ Hypervolume

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17: 29.10 Lamé Functions with Imaginary Periods

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29.10.2 $$z^{\prime }=\mathrm{i}(z-K-\mathrm{i}{K}^{\prime }),$$

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Symbols:

$K^{\prime }\left(k\right)$: Legendre’s complementary complete elliptic integral of the first kind, $K\left(k\right)$: Legendre’s complete elliptic integral of the first kind, $\mathrm{i}$: imaginary unit, $z$: complex variable and $k$: real parameter

Permalink:

http://dlmf.nist.gov/29.10.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §29.10 and Ch.29

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${\mathrm{𝐸𝑐}}_{\nu }^{2m}(\mathrm{i}(z-K-\mathrm{i}{K}^{\prime }),k_{}^{\prime }{}_{}{}^2),$

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${\mathrm{𝐸𝑐}}_{\nu }^{2m+1}(\mathrm{i}(z-K-\mathrm{i}{K}^{\prime }),k_{}^{\prime }{}_{}{}^2),$

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${\mathrm{𝐸𝑠}}_{\nu }^{2m+1}(\mathrm{i}(z-K-\mathrm{i}{K}^{\prime }),k_{}^{\prime }{}_{}{}^2),$

… ►The first and the fourth functions have period $2\mathrm{i}{K}^{\prime }$; the second and the third have period $4\mathrm{i}{K}^{\prime }$. …

18: 19.3 Graphics

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Figure 19.3.1: $K\left(k\right)$ and $E\left(k\right)$ as functions of $k^2$ for $-2\le k^2\le 1$. … Magnify

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Figure 19.3.3: $F(\varphi ,k)$ as a function of $k^2$ and ${\sin }^2\varphi$ for $-1\le k^2\le 2$, $0\le {\sin }^2\varphi \le 1$. If ${\sin }^2\varphi =1$ ($\ge k^2$), then the function reduces to $K\left(k\right)$, becoming infinite when $k^2=1$. If ${\sin }^2\varphi =1/k^2$ (), then it has the value $K\left(1/k\right)/k$: put $c=k^2$ in (19.25.5) and use (19.25.1). Magnify 3D Help

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Figure 19.3.9: $\mathrm{\Re }\left(K\left(k\right)\right)$ as a function of complex $k^2$ for $-2\le \mathrm{\Re }\left(k^2\right)\le 2$, $-2\le \mathrm{\Im }\left(k^2\right)\le 2$. …On the branch cut ($k^2\ge 1$) it is infinite at $k^2=1$, and has the value $K(1/k)/k$ when $k^2>1$. Magnify 3D Help

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Figure 19.3.11: $\mathrm{\Re }\left(E\left(k\right)\right)$ as a function of complex $k^2$ for $-2\le \mathrm{\Re }\left(k^2\right)\le 2$, $-2\le \mathrm{\Im }\left(k^2\right)\le 2$. …On the branch cut ($k^2>1$) it has the value $kE\left(1/k\right)+(k_{}^{\prime }{}_{}{}^2/k)K\left(1/k\right)$, with limit 1 as $k^2\to 1+$. Magnify 3D Help

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19: 29.4 Graphics

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Figure 29.4.9: $a_{\nu }^0\left(k^2\right)$ as a function of $\nu$ and $k^2$. Magnify 3D Help

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Figure 29.4.10: $b_{\nu }^1\left(k^2\right)$ as a function of $\nu$ and $k^2$. Magnify 3D Help

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Figure 29.4.11: $a_{\nu }^1\left(k^2\right)$ as a function of $\nu$ and $k^2$. Magnify 3D Help

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Figure 29.4.12: $b_{\nu }^2\left(k^2\right)$ as a function of $\nu$ and $k^2$. Magnify 3D Help

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Figure 29.4.25: ${\mathrm{𝐸𝑐}}_{1.5}^0(x,k^2)$ as a function of $x$ and $k^2$. Magnify 3D Help

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20: 22.15 Inverse Functions

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22.15.3 $$\mathrm{dn}(\zeta ,k)=x,$$ $k^{\prime }\le x\le 1$,

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Symbols:

$\mathrm{dn}(z,k)$: Jacobian elliptic function, $x$: real, $k$: modulus, $k^{\prime }$: complementary modulus and $\zeta$: solution

Referenced by:

§22.15(i)

Permalink:

http://dlmf.nist.gov/22.15.E3

Encodings:

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See also:

Annotations for §22.15(i), §22.15 and Ch.22

►are denoted respectively by … ►

22.15.5 $$-K\le \mathrm{arcsn}(x,k)\le K,$$

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Symbols:

$K\left(k\right)$: Legendre’s complete elliptic integral of the first kind, $\mathrm{arcsn}(x,k)$: inverse Jacobian elliptic function, $x$: real and $k$: modulus

Permalink:

http://dlmf.nist.gov/22.15.E5

Encodings:

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See also:

Annotations for §22.15(i), §22.15 and Ch.22

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22.15.6 $$0\le \mathrm{arccn}(x,k)\le 2K,$$

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Symbols:

$K\left(k\right)$: Legendre’s complete elliptic integral of the first kind, $\mathrm{arccn}(x,k)$: inverse Jacobian elliptic function, $x$: real and $k$: modulus

Permalink:

http://dlmf.nist.gov/22.15.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §22.15(i), §22.15 and Ch.22

… ►Equations (22.15.1) and (22.15.4), for $\mathrm{arcsn}(x,k)$, are equivalent to (22.15.12) and also to …