# §22.17 Moduli Outside the Interval [0,1]

## §22.17(i) Real or Purely Imaginary Moduli

Jacobian elliptic functions with real moduli in the intervals $(-\infty,0)$ and $(1,\infty)$, or with purely imaginary moduli are related to functions with moduli in the interval $[0,1]$ by the following formulas.

First

 22.17.1 $\operatorname{pq}\left(z,k\right)=\operatorname{pq}\left(z,-k\right),$ ⓘ Symbols: $\operatorname{pq}\left(\NVar{z},\NVar{k}\right)$: generic Jacobian elliptic function, $z$: complex and $k$: modulus Permalink: http://dlmf.nist.gov/22.17.E1 Encodings: TeX, pMML, png See also: Annotations for §22.17(i), §22.17 and Ch.22

for all twelve functions.

Secondly,

 22.17.2 $\displaystyle\operatorname{sn}\left(z,1/k\right)$ $\displaystyle=k\operatorname{sn}\left(z/k,k\right),$ ⓘ Symbols: $\operatorname{sn}\left(\NVar{z},\NVar{k}\right)$: Jacobian elliptic function, $z$: complex and $k$: modulus A&S Ref: 16.11.2 Permalink: http://dlmf.nist.gov/22.17.E2 Encodings: TeX, pMML, png See also: Annotations for §22.17(i), §22.17 and Ch.22 22.17.3 $\displaystyle\operatorname{cn}\left(z,1/k\right)$ $\displaystyle=\operatorname{dn}\left(z/k,k\right),$ ⓘ Symbols: $\operatorname{cn}\left(\NVar{z},\NVar{k}\right)$: Jacobian elliptic function, $\operatorname{dn}\left(\NVar{z},\NVar{k}\right)$: Jacobian elliptic function, $z$: complex and $k$: modulus A&S Ref: 16.11.3 Permalink: http://dlmf.nist.gov/22.17.E3 Encodings: TeX, pMML, png See also: Annotations for §22.17(i), §22.17 and Ch.22 22.17.4 $\displaystyle\operatorname{dn}\left(z,1/k\right)$ $\displaystyle=\operatorname{cn}\left(z/k,k\right).$ ⓘ Symbols: $\operatorname{cn}\left(\NVar{z},\NVar{k}\right)$: Jacobian elliptic function, $\operatorname{dn}\left(\NVar{z},\NVar{k}\right)$: Jacobian elliptic function, $z$: complex and $k$: modulus A&S Ref: 16.11.4 Permalink: http://dlmf.nist.gov/22.17.E4 Encodings: TeX, pMML, png See also: Annotations for §22.17(i), §22.17 and Ch.22

Thirdly, with

 22.17.5 $\displaystyle k_{1}$ $\displaystyle=\frac{k}{\sqrt{1+k^{2}}},$ $\displaystyle k_{1}k_{1}^{\prime}$ $\displaystyle=\frac{k}{1+k^{2}},$ ⓘ Defines: $k_{1}$: change of variable (locally) and $k_{1}^{\prime}$: change of variable (locally) Symbols: $k$: modulus A&S Ref: 16.10.1 (modified) Referenced by: §22.17(i) Permalink: http://dlmf.nist.gov/22.17.E5 Encodings: TeX, TeX, pMML, pMML, png, png See also: Annotations for §22.17(i), §22.17 and Ch.22
 22.17.6 $\displaystyle\operatorname{sn}\left(z,ik\right)$ $\displaystyle=k_{1}^{\prime}\operatorname{sd}\left(z/k_{1}^{\prime},k_{1}% \right),$ 22.17.7 $\displaystyle\operatorname{cn}\left(z,ik\right)$ $\displaystyle=\operatorname{cd}\left(z/k_{1}^{\prime},k_{1}\right),$ 22.17.8 $\displaystyle\operatorname{dn}\left(z,ik\right)$ $\displaystyle=\operatorname{nd}\left(z/k_{1}^{\prime},k_{1}\right).$

In terms of the coefficients of the power series of §22.10(i), the above equations are polynomial identities in $k$. In (22.17.5) either value of the square root can be chosen.

## §22.17(ii) Complex Moduli

When $z$ is fixed each of the twelve Jacobian elliptic functions is a meromorphic function of $k^{2}$. For illustrations see Figures 22.3.2522.3.29. In consequence, the formulas in this chapter remain valid when $k$ is complex. In particular, the Landen transformations in §§22.7(i) and 22.7(ii) are valid for all complex values of $k$, irrespective of which values of $\sqrt{k}$ and $k^{\prime}=\sqrt{1-k^{2}}$ are chosen—as long as they are used consistently. For proofs of these results and further information see Walker (2003).