Editing Jacobi elliptic functions (section)

==Notation==

The elliptic functions can be given in a variety of notations, which can make the subject unnecessarily confusing.  Elliptic functions are functions of two variables.  The first variable might be given in terms of the '''amplitude''' <math>\varphi</math>, or more commonly, in terms of <math>u</math> given below. The second variable might be given in terms of the '''parameter''' <math>m</math>,  or as the '''[[elliptic modulus]]''' <math>k</math>, where <math>k^2=m</math>, or in terms of the '''[[modular angle]]''' <math>\alpha</math>, where <math>m=\sin^2\alpha</math>. The complements of <math>k</math> and <math>m</math> are defined as <math>m'=1-m</math> and <math display="inline">k' = \sqrt{m'}</math>. These four terms are used below without comment to simplify various expressions.

The twelve Jacobi elliptic functions are generally written as <math>\operatorname{pq}(u, m)</math> where <math>\mathrm p</math> and <math>\mathrm q</math> are any of the letters <math>\mathrm c</math>, <math>\mathrm s</math>, <math>\mathrm n</math>, and <math>\mathrm d</math>. Functions of the form <math>\operatorname{pp}(u,m)</math> are trivially set to unity for notational completeness. The “major” functions are generally taken to be <math>\operatorname{cn}(u,m)</math>, <math>\operatorname{sn}(u,m)</math> and <math>\operatorname{dn}(u,m)</math> from which all other functions can be derived and expressions are often written solely in terms of these three functions, however, various symmetries and generalizations are often most conveniently expressed using the full set. (This notation is due to [[Christof Gudermann|Gudermann]] and [[James Whitbread Lee Glaisher|Glaisher]] and is not Jacobi's original notation.)

Throughout this article, <math>\operatorname{pq}(u,t^2)=\operatorname{pq}(u;t)</math>.

The functions are notationally related to each other by the multiplication rule: (arguments suppressed)

:<math>\operatorname{pq}\cdot \operatorname{p'q'}= \operatorname{pq'}\cdot \operatorname{p'q}</math>

from which other commonly used relationships can be derived:

:<math>\frac{\operatorname{pr}}{\operatorname{qr}}=\operatorname{pq}</math>
:<math>\operatorname{pr}\cdot \operatorname{rq}=\operatorname{pq}</math>
:<math>\frac{1}{\operatorname{qp}}=\operatorname{pq}</math>

The multiplication rule follows immediately from the identification of the elliptic functions with the [[Neville theta function]]s<ref name="Neville1944">{{cite book |last=Neville |first=Eric Harold |date=1944 |title=Jacobian Elliptic Functions |url=https://archive.org/details/jacobianelliptic00neviuoft |location=Oxford |publisher=Oxford University Press |author-link=Eric Harold Neville}}</ref>

:<math>\operatorname{pq}(u,m)=\frac{\theta_\operatorname{p}(u,m)}{\theta_\operatorname{q}(u,m)}</math>
Also note that:

: <math>K(m)=K(k^2)=\int_0^1\frac{dt}{\sqrt{(1-t^2)(1-mt^2)}}=\int_0^1\frac{dt}{\sqrt{(1-t^2)(1-k^2t^2)}}. </math>