Jacobi elliptic functions: Difference between revisions

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The value of the Jacobi transformations is that any set of Jacobi elliptic functions with any complexreal-valued parameter ''m'' can be converted into another set for which <math>0&nbsp;≤&nbsp;''<m''&nbsp;≤&nbsp;\le 1/2</math> and, for real values of ''u'', the function values will be real.<ref name="Neville1944"/>{{rp|p. 215}}

===Amplitude transformations===

:<math>\operatorname{am}\left(\sqrt{m'}u,-\frac{m}{m'}\right)=\frac{\pi}{2}-\operatorname{am}(K-u,m),\quad u\in\mathbb{R},\, 0<m<1,</math>

:<math>\operatorname{am}(u,m')=-2\arctan\left(i\tan \frac{\operatorname{am}(iu,m)}{2}\right),\quad \left|\operatorname{Re}u\right|<K',\, \left|\operatorname{Im}u\right|<K,\, 0<m<1,</math>

:<math>\operatorname{am}\left(\sqrt{m}u,\frac{1}{m}\right)=\arcsin(\sqrt{m}\operatorname{sn}(u,m)),\quad -2K<u<2K,\, 0<m<1</math>

==The Jacobi hyperbola==

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Let the [[nome (mathematics)|nome]] be <math>q=\exp(-\pi K'(m)/K(m))=e^{i\pi \tau}</math>, <math>\operatorname{Im}(\tau)>0</math>, <math>m=k^2</math> and let <math>v=\pi u /(2K(m))</math>. Then the functions have expansions as [[Lambert series]]

:<math>\operatorname{am}(u,m)=\frac{\pi u}{2K(m)}+2\sum_{n=1}^\infty \frac{q^n}{n(1+q^{2n})}\sin (2nv),</math>

:<math>\operatorname{sn}(u,m)=\frac{2\pi}{kK(m)}

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:<math>\operatorname{dn}(u,m)=\frac{\pi}{2K(m)} + \frac{2\pi}{K(m)}

\sum_{n=1}^\infty \frac{q^{n}}{1+q^{2n}} \cos (2nv),</math>

:<math>\operatorname{zn}(u,m)=\frac{2\pi}{K(m)}\sum_{n=1}^\infty \frac{q^n}{1-q^{2n}}\sin (2nv)</math>

when <math>q\exp (2\left|\operatorname{Im} (vu/K)\right|<\operatorname{Im}(iK'/K)<1.</math>.

For the Jacobi amplitude,

:<math>\operatorname{am}(u,m)=\frac{\pi u}{2K(m)}+2\sum_{n=1}^\infty \frac{q^n}{n(1+q^{2n})}\sin (2nv)</math>

where <math>0<m<1</math> and <math>u\in\mathbb{R}</math>.

Bivariate power series expansions have been published by Schett.<ref>{{cite journal|first1=Alois|last1=Schett |title=Properties of the Taylor series expansion coefficients of the Jacobian Elliptic Functions|year=1976|journal=Math. Comp.|volume=30|number=133|pages=143–147|doi=10.1090/S0025-5718-1976-0391477-3| mr=0391477|s2cid=120666361 }}</ref>