§22.5 Special Values

Keywords:: Jacobian elliptic functions
Permalink:: http://dlmf.nist.gov/22.5

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

§22.5(i) Special Values of $z$

Notes:: See Lawden (1989, pp. 26–30), Whittaker and Watson (1927, pp. 498–505).
Keywords:: Jacobian elliptic functions
Permalink:: http://dlmf.nist.gov/22.5.i

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 $z=K+iK^{{\prime}}$ , $\mathop{\mathrm{sn}\/}\nolimits\left(z,k\right)=1/k$ , $\ifrac{d\mathop{\mathrm{sn}\/}\nolimits\left(z,k\right)}{dz}=0$ . (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	$K$	$K+iK^{{\prime}}$	$iK^{{\prime}}$	$2K$	$2K+2iK^{{\prime}}$	$2iK^{{\prime}}$
$\mathop{\mathrm{sn}\/}\nolimits z$	$0,1$	$1,0$	$1/k,0$	$\infty$ , $1/k$	$0,-1$	$0,-1$	$0,1$
$\mathop{\mathrm{cn}\/}\nolimits z$	$1,0$	$0,-k^{{\prime}}$	$-ik^{{\prime}}/k,0$	$\infty$ , $-i/k$	$-1,0$	$1,0$	$-1,0$
$\mathop{\mathrm{dn}\/}\nolimits z$	$1,0$	$k^{{\prime}},0$	$0,ik^{{\prime}}$	$\infty$ , $-i$	$1,0$	$-1,0$	$-1,0$
$\mathop{\mathrm{cd}\/}\nolimits z$	$1,0$	$0,-1$	$\infty,-k^{{-1}}$	$k^{{-1}},0$	$-1,0$	$-1,0$	$1,0$
$\mathop{\mathrm{sd}\/}\nolimits z$	$0,1$	${k^{{\prime}}}^{{-1}},0$	$\infty,-i(kk^{{\prime}})^{{-1}}$	$ik^{{-1}},0$	$0,-1$	$0,1$	$0,-1$
$\mathop{\mathrm{nd}\/}\nolimits z$	$1,0$	${k^{{\prime}}}^{{-1}},0$	$\infty,-i{k^{{\prime}}}^{{-1}}$	$0,i$	$1,0$	$-1,0$	$-1,0$
$\mathop{\mathrm{dc}\/}\nolimits z$	$1,0$	$\infty,-1$	$0,k$	$k,0$	$-1,0$	$-1,0$	$1,0$
$\mathop{\mathrm{nc}\/}\nolimits z$	$1,0$	$\infty,-{k^{{\prime}}}^{{-1}}$	$ik{k^{{\prime}}}^{{-1}},0$	$0,ik$	$-1,0$	$1,0$	$-1,0$
$\mathop{\mathrm{sc}\/}\nolimits z$	$0,1$	$\infty,-{k^{{\prime}}}^{{-1}}$	$i{k^{{\prime}}}^{{-1}},0$	$i,0$	$0,1$	$0,-1$	$0,-1$
$\mathop{\mathrm{ns}\/}\nolimits z$	$\infty,1$	$1,0$	$k,0$	$0,k$	$\infty,-1$	$\infty,-1$	$\infty,1$
$\mathop{\mathrm{ds}\/}\nolimits z$	$\infty,1$	$k^{{\prime}},0$	$0,ikk^{{\prime}}$	$-ik,0$	$\infty,-1$	$\infty,1$	$\infty,-1$
$\mathop{\mathrm{cs}\/}\nolimits z$	$\infty,1$	$0,-k^{{\prime}}$	$-ik^{{\prime}},0$	$-i,0$	$\infty,1$	$\infty,-1$	$\infty,-1$

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

Table 22.5.2: Other special values of Jacobian elliptic functions.

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

Symbols:: $\mathop{\mathrm{cn}\/}\nolimits\left(z,k\right)$ : Jacobian elliptic function, $\mathop{\mathrm{dn}\/}\nolimits\left(z,k\right)$ : Jacobian elliptic function, $\mathop{\mathrm{sn}\/}\nolimits\left(z,k\right)$ : Jacobian elliptic function, $\mathop{{K^{{\prime}}}\/}\nolimits\!\left(k\right)$ : Legendre’s complementary complete elliptic integral of the first kind, $\mathop{K\/}\nolimits\!\left(k\right)$ : Legendre’s complete elliptic integral of the first kind, $z$ : complex, $k$ : modulus and $k^{{\prime}}$ : complementary modulus
A&S Ref:: Table 16.5
Referenced by:: §22.5(i)
Permalink:: http://dlmf.nist.gov/22.5.T2

§22.5(ii) Limiting Values of $k$

Notes:: See Lawden (1989, pp. 24–26 and 38–40).
Keywords:: Jacobian elliptic functions
Permalink:: http://dlmf.nist.gov/22.5.ii

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 $k\to 0$ .

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

Table 22.5.4: Limiting forms of Jacobian elliptic functions as $k\to 1$ .

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

Symbols:: $\mathop{\mathrm{cd}\/}\nolimits\left(z,k\right)$ : Jacobian elliptic function, $\mathop{\mathrm{cn}\/}\nolimits\left(z,k\right)$ : Jacobian elliptic function, $\mathop{\mathrm{cs}\/}\nolimits\left(z,k\right)$ : Jacobian elliptic function, $\mathop{\mathrm{dc}\/}\nolimits\left(z,k\right)$ : Jacobian elliptic function, $\mathop{\mathrm{dn}\/}\nolimits\left(z,k\right)$ : Jacobian elliptic function, $\mathop{\mathrm{ds}\/}\nolimits\left(z,k\right)$ : Jacobian elliptic function, $\mathop{\mathrm{nc}\/}\nolimits\left(z,k\right)$ : Jacobian elliptic function, $\mathop{\mathrm{nd}\/}\nolimits\left(z,k\right)$ : Jacobian elliptic function, $\mathop{\mathrm{ns}\/}\nolimits\left(z,k\right)$ : Jacobian elliptic function, $\mathop{\mathrm{sc}\/}\nolimits\left(z,k\right)$ : Jacobian elliptic function, $\mathop{\mathrm{sd}\/}\nolimits\left(z,k\right)$ : Jacobian elliptic function, $\mathop{\mathrm{sn}\/}\nolimits\left(z,k\right)$ : Jacobian elliptic function, $\mathop{\mathrm{csch}\/}\nolimits z$ : hyperbolic cosecant function, $\mathop{\cosh\/}\nolimits z$ : hyperbolic cosine function, $\mathop{\coth\/}\nolimits z$ : hyperbolic cotangent function, $\mathop{\mathrm{sech}\/}\nolimits z$ : hyperbolic secant function, $\mathop{\sinh\/}\nolimits z$ : hyperbolic sine function, $\mathop{\tanh\/}\nolimits z$ : hyperbolic tangent function, $z$ : complex and $k$ : modulus
A&S Ref:: Table 16.6
Referenced by:: §22.5(ii), Version 1.0.1 (June 27, 2011)
Permalink:: http://dlmf.nist.gov/22.5.T4
Errata (effective with 1.0.1):: Originally, the limiting form for $\mathop{\mathrm{sc}\/}\nolimits\left(z,k\right)$ was incorrect; it was given as $\mathop{\cosh\/}\nolimits z$ , instead of $\mathop{\sinh\/}\nolimits z$ .
Reported 2010-11-23

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 $k^{2}=e^{{i\pi/3}}$ (equianharmonic case) see §23.5(v).

§22.5 Special Values

Contents

§22.5(i) Special Values of

§22.5(ii) Limiting Values of

§22.5(i) Special Values of $z$

§22.5(ii) Limiting Values of $k$