Editing Jacobi elliptic functions (section)

==The Jacobi hyperbola==
[[File:Jacobi Elliptic Functions (on Jacobi Hyperbola).svg|right|thumb|upright=1.5|Plot of the Jacobi hyperbola (''x''<sup>2</sup>&nbsp;+&nbsp;''y''<sup>2</sup>/''b''<sup>2</sup>&nbsp;=&nbsp;1, ''b'' imaginary) and the twelve Jacobi Elliptic functions pq(''u'',''m'') for particular values of angle ''&phi;'' and  parameter ''b''.  The solid curve is the hyperbola, with  ''m''&nbsp;=&nbsp;1&nbsp;−&nbsp;1/''b''<sup>2</sup> and ''u''&nbsp;=&nbsp;''F''(''&phi;'',''m'') where ''F''(&sdot;,&sdot;) is the [[elliptic integral]] of the first kind. The dotted curve is the unit circle. For the ds-dc triangle, ''&sigma;''&nbsp;=&nbsp; sin(''&phi;'')cos(''&phi;'').]]

Introducing complex numbers, our ellipse has an associated hyperbola:

:<math> x^2 - \frac{y^2}{b^2} = 1 </math> 
from applying Jacobi's imaginary transformation<ref name="WolframJE"/> to the elliptic functions in the above equation for ''x'' and&nbsp;''y''.

:<math> x = \frac{1} {\operatorname{dn}(u,1-m)},\quad y = \frac{ \operatorname{sn}(u,1-m)} {\operatorname{dn}(u,1-m)}</math>

It follows that we can put <math> x=\operatorname{dn}(u,1-m), y=\operatorname{sn}(u,1-m)</math>. So our ellipse has a dual ellipse with m replaced by 1-m. This leads to the complex torus mentioned in the  Introduction.<ref>{{Cite web|url=https://paramanands.blogspot.co.uk/2011/01/elliptic-functions-complex-variables.html#.WlHhTbp2t9A|title = Elliptic Functions: Complex Variables}}</ref> Generally, m may be a complex number, but when m is real and m<0, the curve is an ellipse with major axis in the x direction. At m=0 the curve is a circle, and for 0<m<1, the curve is an ellipse with major axis in the y direction. At ''m''&nbsp;=&nbsp;1, the curve degenerates into two vertical lines at ''x''&nbsp;=&nbsp;±1. For ''m''&nbsp;>&nbsp;1, the curve is a hyperbola. When ''m'' is complex but not real, ''x'' or ''y'' or both are complex and the curve cannot be described on a real ''x''-''y'' diagram.