Editing Jacobi elliptic functions (section)

=={{anchor|sn|cn|dn|am}}Definition in terms of inverses of elliptic integrals==

[[File:Modell der elliptischen Funktion φ=am (u, k) durch eine Fläche -Schilling V, 1 - 317-.jpg|thumb|Model of the Jacobi amplitude (measured along vertical axis) as a function of independent variables ''u'' and the modulus ''k'']]

There is a definition, relating the elliptic functions to the inverse of the [[Elliptic integral#Incomplete elliptic integral of the first kind|incomplete elliptic integral of the first kind]] <math>F</math>.  These functions take the parameters <math>u</math> and <math>m</math> as inputs. The <math>\varphi</math> that satisfies

:<math>u=F(\varphi,m)=\int_0^\varphi \frac{\mathrm d\theta} {\sqrt {1-m \sin^2 \theta}}</math>

is called the '''Jacobi amplitude''':

:<math>\operatorname{am}(u,m)=\varphi.</math>

In this framework, the ''elliptic sine'' sn&nbsp;''u'' (Latin: ''sinus amplitudinis'') is given by

:<math>\operatorname {sn} (u,m) = \sin \operatorname{am}(u,m)</math>

and the ''elliptic cosine'' cn&nbsp;''u'' (Latin: ''cosinus amplitudinis'') is given by

:<math>\operatorname {cn} (u,m) = \cos \operatorname{am}(u,m)</math>

and the ''delta amplitude'' dn&nbsp;''u'' (Latin: ''delta amplitudinis'')<ref group="note">If <math>u\in\mathbb{R}</math> and <math>m</math> is restricted to <math>[0,1]</math>, then <math>\operatorname{dn}(u,m)</math> can be also written as <math>\sqrt {1-m\sin^2 \operatorname{am}(u,m)}.</math></ref>

:<math>\operatorname {dn} (u,m) = \frac{\mathrm d}{\mathrm du}\operatorname{am}(u,m).</math>
In the above, the value <math>m</math> is a free parameter, usually taken to be real such that <math>0\leq m \leq 1</math> (but can be complex in general), and so the elliptic functions can be thought of as being given by two variables, <math>u</math> and the parameter&nbsp;<math>m</math>. The remaining nine elliptic functions are easily built from the above three (<math>\operatorname{sn}</math>, <math>\operatorname{cn}</math>, <math>\operatorname{dn}</math>), and are given in a section below. Note that when <math>\varphi=\pi/2</math>, that <math>u</math> then equals the [[quarter period]]&nbsp;<math>K</math>.

In the most general setting, <math>\operatorname{am}(u,m)</math> is a [[multivalued function]] (in <math>u</math>) with infinitely many [[Branch point|logarithmic branch points]] (the branches differ by integer multiples of <math>2\pi</math>), namely the points <math>2sK(m)+(4t+1)K(1-m)i</math> and <math>2sK(m)+(4t+3)K(1-m)i</math> where <math>s,t\in\mathbb{Z}</math>.<ref name="sala">{{cite journal |last=Sala |first=Kenneth L. |date=November 1989 |title=Transformations of the Jacobian Amplitude Function and Its Calculation via the Arithmetic-Geometric Mean|url=https://epubs.siam.org/doi/abs/10.1137/0520100 |journal=SIAM Journal on Mathematical Analysis|volume=20|issue=6|pages=1514–1528|doi=10.1137/0520100 }}</ref> This multivalued function can be made single-valued by cutting the complex plane along the line segments joining these branch points (the cutting can be done in non-equivalent ways, giving non-equivalent single-valued functions), thus making <math>\operatorname{am}(u,m)</math> [[Analytic function|analytic]] everywhere except on the [[Branch point#Branch cuts|branch cuts]]. In contrast, <math>\sin\operatorname{am}(u,m)</math> and other elliptic functions have no branch points, give consistent values for every branch of <math>\operatorname{am}</math>, and are [[meromorphic function|meromorphic]] in the whole complex plane. Since every elliptic function is meromorphic in the whole complex plane (by definition), <math>\operatorname{am}(u,m)</math> (when considered as a single-valued function) is not an elliptic function.

However, a particular cutting for <math>\operatorname{am}(u,m)</math> can be made in the <math>u</math>-plane by line segments from <math>2sK(m)+(4t+1)K (1-m)i</math> to <math>2sK(m)+(4t+3)K(1-m)i</math> with <math>s,t\in\mathbb{Z}</math>; then it only remains to define <math>\operatorname{am}(u,m)</math> at the branch cuts by continuity from some direction. Then <math>\operatorname{am}(u,m)</math> becomes single-valued and singly-periodic in <math>u</math> with the minimal period <math>4iK(1-m)</math> and it has singularities at the logarithmic branch points mentioned above. If <math>m\in\mathbb{R}</math> and <math>m\le 1</math>, <math>\operatorname{am}(u,m)</math> is continuous in <math>u</math> on the real line. When <math>m>1</math>, the branch cuts of <math>\operatorname{am}(u,m)</math> in the <math>u</math>-plane cross the real line at <math>2(2s+1)K(1/m)/\sqrt{m}</math> for <math>s\in\mathbb{Z}</math>; therefore for <math>m>1</math>, <math>\operatorname{am}(u,m)</math> is not continuous in <math>u</math> on the real line and jumps by <math>2\pi</math> on the discontinuities.

But defining <math>\operatorname{am}(u,m)</math> this way gives rise to very complicated branch cuts in the <math>m</math>-plane (''not'' the <math>u</math>-plane); they have not been fully described as of yet.

Let
:<math>E(\varphi,m)=\int_0^{\varphi}\sqrt{1-m\sin^2\theta}\,\mathrm d\theta</math>
be the [[Elliptic integral#Incomplete elliptic integral of the second kind|incomplete elliptic integral of the second kind]] with parameter <math>m</math>.

Then the '''Jacobi epsilon''' function can be defined as
:<math>\mathcal{E}(u,m)=E(\operatorname{am}(u,m),m)</math>
for <math>u\in\mathbb{R}</math> and <math>0<m<1</math> and by [[analytic continuation]] in each of the variables otherwise: the Jacobi epsilon function is meromorphic in the whole complex plane (in both <math>u</math> and <math>m</math>). Alternatively, throughout both the <math>u</math>-plane and <math>m</math>-plane,<ref>{{dlmf|first1=W. P.|last1=Reinhardt|first2=P. L.|last2=Walker|id=22.16.E17|title=Jacobian Elliptic Functions}}</ref>
:<math>\mathcal{E} (u,m)=\int_0^u \operatorname{dn}^2(t,m)\, \mathrm dt;</math>
<math>\mathcal{E}</math> is well-defined in this way because all [[Residue (complex analysis)|residues]] of <math>t\mapsto\operatorname{dn}(t,m)^2</math> are zero, so the integral is path-independent. So the Jacobi epsilon relates the incomplete elliptic integral of the first kind to the incomplete elliptic integral of the second kind:
:<math>E(\varphi,m)=\mathcal{E}(F(\varphi,m),m).</math>
The Jacobi epsilon function is not an elliptic function, but it appears when differentiating the Jacobi elliptic functions with respect to the parameter.

The '''Jacobi zn''' function is defined by
:<math>\operatorname{zn}(u,m)=\mathcal{E}(u,m)-\frac{E(m)}{K(m)}u.</math>
It is a singly periodic function which is meromorphic in <math>u</math>, but not in <math>m</math> (due to the branch cuts of <math>E</math> and <math>K</math>). Its minimal period in <math>u</math> is <math>2K(m)</math>. It is related to the [[Elliptic integrals#Jacobi zeta function|Jacobi zeta function]] by <math>Z(\varphi,m)=\operatorname{zn}(F(\varphi,m),m).</math>

Historically, the Jacobi elliptic functions were first defined by using the amplitude. In more modern texts on elliptic functions, the Jacobi elliptic functions are defined by other means, for example by ratios of theta functions (see below), and the amplitude is ignored.

In modern terms, the relation to elliptic integrals would be expressed by <math>\operatorname{sn}(F(\varphi,m),m)=\sin\varphi</math> (or <math>\operatorname{cn}(F(\varphi,m),m)=\cos\varphi</math>) instead of <math>\operatorname{am}(F(\varphi,m),m)=\varphi</math>.