22 Jacobian Elliptic Functions Properties 22.4 Periods, Poles, and Zeros 22.6 Elementary Identities

§22.5 Special Values

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Keywords:: Jacobian elliptic functions, special values of the variable
Permalink:: http://dlmf.nist.gov/22.5
See also:: Annotations for Ch.22

§22.5(i) Special Values of $z$
§22.5(ii) Limiting Values of $k$

§22.5(i) Special Values of $z$

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Keywords:: Jacobian elliptic functions, special values of the variable
Notes:: See Lawden (1989, pp. 26–30), Whittaker and Watson (1927, pp. 498–505).
Permalink:: http://dlmf.nist.gov/22.5.i
See also:: Annotations for §22.5 and Ch.22

Table 22.5.1 gives the value of each of the 12 Jacobian elliptic functions, together with its $z$ -derivative (or at a pole, the residue), for values of $z$ that are integer multiples of $K$ , $iK^{\prime}$ . For example, at $z=K+iK^{\prime}$ , $\operatorname{sn}\left(z,k\right)=1/k$ , $\ifrac{\mathrm{d}\operatorname{sn}\left(z,k\right)}{\mathrm{d}z}=0$ . (The modulus $k$ is suppressed throughout the table.)

Table 22.5.1: Jacobian elliptic function values, together with derivatives or residues, for special values of the variable.

	$z$
	$0$	$K$	$K+iK^{\prime}$	$iK^{\prime}$	$2K$	$2K+2iK^{\prime}$	$2iK^{\prime}$
$\operatorname{sn}z$	$0,1$	$1,0$	$1/k,0$	$\infty$ , $1/k$	$0,-1$	$0,-1$	$0,1$
$\operatorname{cn}z$	$1,0$	$0,-k^{\prime}$	$-ik^{\prime}/k,0$	$\infty$ , $-i/k$	$-1,0$	$1,0$	$-1,0$
$\operatorname{dn}z$	$1,0$	$k^{\prime},0$	$0,ik^{\prime}$	$\infty$ , $-i$	$1,0$	$-1,0$	$-1,0$
$\operatorname{cd}z$	$1,0$	$0,-1$	$\infty,-k^{-1}$	$k^{-1},0$	$-1,0$	$-1,0$	$1,0$
$\operatorname{sd}z$	$0,1$	${k^{\prime}}^{-1},0$	$\infty,-i(kk^{\prime})^{-1}$	$ik^{-1},0$	$0,-1$	$0,1$	$0,-1$
$\operatorname{nd}z$	$1,0$	${k^{\prime}}^{-1},0$	$\infty,-i{k^{\prime}}^{-1}$	$0,i$	$1,0$	$-1,0$	$-1,0$
$\operatorname{dc}z$	$1,0$	$\infty,-1$	$0,k$	$k,0$	$-1,0$	$-1,0$	$1,0$
$\operatorname{nc}z$	$1,0$	$\infty,-{k^{\prime}}^{-1}$	$ik{k^{\prime}}^{-1},0$	$0,ik$	$-1,0$	$1,0$	$-1,0$
$\operatorname{sc}z$	$0,1$	$\infty,-{k^{\prime}}^{-1}$	$i{k^{\prime}}^{-1},0$	$i,0$	$0,1$	$0,-1$	$0,-1$
$\operatorname{ns}z$	$\infty,1$	$1,0$	$k,0$	$0,k$	$\infty,-1$	$\infty,-1$	$\infty,1$
$\operatorname{ds}z$	$\infty,1$	$k^{\prime},0$	$0,ikk^{\prime}$	$-ik,0$	$\infty,-1$	$\infty,1$	$\infty,-1$
$\operatorname{cs}z$	$\infty,1$	$0,-k^{\prime}$	$-ik^{\prime},0$	$-i,0$	$\infty,1$	$\infty,-1$	$\infty,-1$

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Symbols:: $\operatorname{cd}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $\operatorname{cn}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $\operatorname{cs}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $\operatorname{dc}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $\operatorname{dn}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $\operatorname{ds}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $\operatorname{nc}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $\operatorname{nd}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $\operatorname{ns}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $\operatorname{sc}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $\operatorname{sd}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $\operatorname{sn}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, ${K^{\prime}}\left(\NVar{k}\right)$ : Legendre’s complementary complete elliptic integral of the first kind, $K\left(\NVar{k}\right)$ : Legendre’s complete elliptic integral of the first kind, $\mathrm{i}$ : imaginary unit, $z$ : complex, $k$ : modulus and $k^{\prime}$ : complementary modulus
A&S Ref:: Table 16.7
Referenced by:: §22.20(ii), §22.5(i)
Permalink:: http://dlmf.nist.gov/22.5.T1
See also:: Annotations for §22.5(i), §22.5 and Ch.22

Table 22.5.2 gives $\operatorname{sn}\left(z,k\right)$ , $\operatorname{cn}\left(z,k\right)$ , $\operatorname{dn}\left(z,k\right)$ for other special values of $z$ . For example, $\operatorname{sn}\left(\frac{1}{2}K,k\right)=(1+k^{\prime})^{-1/2}$ . For the other nine functions ratios can be taken; compare (22.2.10).

Table 22.5.2: Other special values of Jacobian elliptic functions.

	$z$
	$\frac{1}{2}K$	$\frac{1}{2}(K+iK^{\prime})$	$\frac{1}{2}iK^{\prime}$
$\operatorname{sn}z$	$(1+k^{\prime})^{-1/2}$	$\ifrac{\left((1+k)^{1/2}+i(1-k)^{1/2}\right)}{(2k)^{1/2}}$	$ik^{-1/2}$
$\operatorname{cn}z$	$(\ifrac{k^{\prime}}{(1+k^{\prime})})^{1/2}$	$\ifrac{(1-i){k^{\prime}}^{1/2}}{(2k)^{1/2}}$	$(1+k)^{1/2}k^{-1/2}$
$\operatorname{dn}z$	${k^{\prime}}^{1/2}$	$\frac{1-i}{2}{k^{\prime}}^{1/2}\left((1+k)^{1/2}+i(1-k)^{1/2}\right)$	$(1+k)^{1/2}$
	$z$
	$\frac{3}{2}K$	$\frac{3}{2}(K+iK^{\prime})$	$\frac{3}{2}iK^{\prime}$
$\operatorname{sn}z$	$(1+k^{\prime})^{-1/2}$	$\ifrac{\left((1+k)^{1/2}+i(1-k)^{1/2}\right)}{(2k)^{1/2}}$	$-ik^{-1/2}$
$\operatorname{cn}z$	$-(\ifrac{k^{\prime}}{(1+k^{\prime})})^{1/2}$	$\ifrac{(1-i){k^{\prime}}^{1/2}}{(2k)^{1/2}}$	$-(1+k)^{1/2}k^{-1/2}$
$\operatorname{dn}z$	${k^{\prime}}^{1/2}$	$\frac{i-1}{2}{k^{\prime}}^{1/2}\left((1+k)^{1/2}+i(1-k)^{1/2}\right)$	$-(1+k)^{1/2}$

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Symbols:: $\operatorname{cn}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $\operatorname{dn}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $\operatorname{sn}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $\pi$ : the ratio of the circumference of a circle to its diameter, ${K^{\prime}}\left(\NVar{k}\right)$ : Legendre’s complementary complete elliptic integral of the first kind, $K\left(\NVar{k}\right)$ : Legendre’s complete elliptic integral of the first kind, $\mathrm{i}$ : imaginary unit, $\operatorname{ph}$ : phase, $z$ : complex, $k$ : modulus and $k^{\prime}$ : complementary modulus
A&S Ref:: Table 16.5
Referenced by:: §22.5(i), Erratum (V1.0.7) for Table 22.5.2, Erratum (V1.1.10) for Table 22.5.2
Permalink:: http://dlmf.nist.gov/22.5.T2
Correction (effective with 1.1.10):: The entry for $\operatorname{sn}z$ , $z=\frac{3}{2}(K+iK^{\prime})$ , which previously was $(1+i)((1+k^{\prime})^{1/2}-i(1-k^{\prime})^{1/2})/(2k^{1/2})$ has been corrected to be $\ifrac{\left((1+k)^{1/2}+i(1-k)^{1/2}\right)}{(2k)^{1/2}}$ . The previous result was correct only for $|\operatorname{ph}k|<\frac{1}{2}\pi$ . Furthermore, the presentation of several of the other results in the middle column have been simplfied.
Suggested 2023-03-06 by Alan Barnes
Errata (effective with 1.0.7):: The entry for $\operatorname{sn}z$ at $z=\frac{3}{2}(K+iK^{\prime})$ has been corrected. The correct entry is $(1+i)((1+k^{\prime})^{1/2}-i(1-k^{\prime})^{1/2})/(2k^{1/2})$ . Originally the terms $(1+k^{\prime})^{1/2}$ and $(1-k^{\prime})^{1/2}$ were given incorrectly as $(1+k)^{1/2}$ and $(1-k)^{1/2}$ . Similarly, the entry for $\operatorname{dn}z$ at $z=\frac{3}{2}(K+iK^{\prime})$ has been corrected. The correct entry is $(-1+i){k^{\prime}}^{1/2}((1+k)^{1/2}+i(1-k)^{1/2})/2$ . Originally the terms $(1+k)^{1/2}$ and $(1-k)^{1/2}$ were given incorrectly as $(1+k^{\prime})^{1/2}$ and $(1-k^{\prime})^{1/2}$ .
Reported 2014-02-28 by Svante Janson
See also:: Annotations for §22.5(i), §22.5 and Ch.22

§22.5(ii) Limiting Values of $k$

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Keywords:: Jacobian elliptic functions, equianharmonic case, lemniscatic case, limiting forms as $k\to 0$ or $k\to 1$ , limiting values, modulus
Notes:: See Lawden (1989, pp. 24–26 and 38–40).
Permalink:: http://dlmf.nist.gov/22.5.ii
See also:: Annotations for §22.5 and Ch.22

If $k\to 0+$ , then $K\to\pi/2$ and $K^{\prime}\to\infty$ ; if $k\to 1-$ , then $K\to\infty$ and $K^{\prime}\to\pi/2$ . In these cases the elliptic functions degenerate into elementary trigonometric and hyperbolic functions, respectively. See Tables 22.5.3 and 22.5.4.

Table 22.5.3: Limiting forms of Jacobian elliptic functions as

k\to 0

$\operatorname{sn}\left(z,k\right)$ $\;\to\;$	$\sin z$	$\operatorname{cd}\left(z,k\right)$ $\;\to\;$	$\cos z$	$\operatorname{dc}\left(z,k\right)$ $\;\to\;$	$\sec z$	$\operatorname{ns}\left(z,k\right)$ $\;\to\;$	$\csc z$
$\operatorname{cn}\left(z,k\right)$ $\;\to\;$	$\cos z$	$\operatorname{sd}\left(z,k\right)$ $\;\to\;$	$\sin z$	$\operatorname{nc}\left(z,k\right)$ $\;\to\;$	$\sec z$	$\operatorname{ds}\left(z,k\right)$ $\;\to\;$	$\csc z$
$\operatorname{dn}\left(z,k\right)$ $\;\to\;$	$1$	$\operatorname{nd}\left(z,k\right)$ $\;\to\;$	$1$	$\operatorname{sc}\left(z,k\right)$ $\;\to\;$	$\tan z$	$\operatorname{cs}\left(z,k\right)$ $\;\to\;$	$\cot z$

Table 22.5.4: Limiting forms of Jacobian elliptic functions as

k\to 1

$\operatorname{sn}\left(z,k\right)$ $\;\to\;$	$\tanh z$	$\operatorname{cd}\left(z,k\right)$ $\;\to\;$	$1$	$\operatorname{dc}\left(z,k\right)$ $\;\to\;$	$1$	$\operatorname{ns}\left(z,k\right)$ $\;\to\;$	$\coth z$
$\operatorname{cn}\left(z,k\right)$ $\;\to\;$	$\operatorname{sech}z$	$\operatorname{sd}\left(z,k\right)$ $\;\to\;$	$\sinh z$	$\operatorname{nc}\left(z,k\right)$ $\;\to\;$	$\cosh z$	$\operatorname{ds}\left(z,k\right)$ $\;\to\;$	$\operatorname{csch}z$
$\operatorname{dn}\left(z,k\right)$ $\;\to\;$	$\operatorname{sech}z$	$\operatorname{nd}\left(z,k\right)$ $\;\to\;$	$\cosh z$	$\operatorname{sc}\left(z,k\right)$ $\;\to\;$	$\sinh z$	$\operatorname{cs}\left(z,k\right)$ $\;\to\;$	$\operatorname{csch}z$

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Symbols:: $\operatorname{cd}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $\operatorname{cn}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $\operatorname{cs}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $\operatorname{dc}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $\operatorname{dn}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $\operatorname{ds}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $\operatorname{nc}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $\operatorname{nd}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $\operatorname{ns}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $\operatorname{sc}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $\operatorname{sd}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $\operatorname{sn}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $\operatorname{csch}\NVar{z}$ : hyperbolic cosecant function, $\cosh\NVar{z}$ : hyperbolic cosine function, $\coth\NVar{z}$ : hyperbolic cotangent function, $\operatorname{sech}\NVar{z}$ : hyperbolic secant function, $\sinh\NVar{z}$ : hyperbolic sine function, $\tanh\NVar{z}$ : hyperbolic tangent function, $z$ : complex and $k$ : modulus
A&S Ref:: Table 16.6
Referenced by:: §22.5(ii), Erratum (V1.0.1) for Table 22.5.4
Permalink:: http://dlmf.nist.gov/22.5.T4
Errata (effective with 1.0.1):: Originally, the limiting form for $\operatorname{sc}\left(z,k\right)$ was incorrect; it was given as $\cosh z$ , instead of $\sinh z$ .
See also:: Annotations for §22.5(ii), §22.5 and Ch.22

Expansions for $K,K^{\prime}$ as $k\to 0$ or $1$ are given in §§19.5, 19.12.

For values of $K,K^{\prime}$ when $k^{2}=\frac{1}{2}$ (lemniscatic case) see §23.5(iii), and for $k^{2}=e^{\mathrm{i}\pi/3}$ (equianharmonic case) see §23.5(v).

22.4 Periods, Poles, and Zeros 22.6 Elementary Identities

NIST

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§22.5 Special Values

Contents

§22.5(i) Special Values of z

§22.5(ii) Limiting Values of k

§22.5(i) Special Values of $z$

§22.5(ii) Limiting Values of $k$