22 Jacobian Elliptic FunctionsApplications22.17 Moduli Outside the Interval [0,1]22.19 Physical Applications

- §22.18(i) Lengths and Parametrization of Plane Curves
- §22.18(ii) Conformal Mapping
- §22.18(iii) Uniformization and Other Parametrizations
- §22.18(iv) Elliptic Curves and the Jacobi–Abel Addition Theorem

22.18.1 | $$\left({x}^{2}/{a}^{2}\right)+\left({y}^{2}/{b}^{2}\right)=1,$$ | ||

with $a\ge b>0$, is parametrized by

22.18.2 | $x$ | $=a\mathrm{sn}(u,k),$ | ||

$y$ | $=b\mathrm{cn}(u,k),$ | |||

where $k=\sqrt{1-({b}^{2}/{a}^{2})}$ is the eccentricity, and $0\le u\le 4K\left(k\right)$. The arc length $l(u)$ in the first quadrant, measured from $u=0$, is

22.18.3 | $$l(u)=a\mathcal{E}(u,k),$$ | ||

where $\mathcal{E}(u,k)$ is Jacobi’s epsilon function (§22.16(ii)).

In polar coordinates, $x=r\mathrm{cos}\varphi $, $y=r\mathrm{sin}\varphi $, the lemniscate is given by ${r}^{2}=\mathrm{cos}\left(2\varphi \right)$, $0\le \varphi \le 2\pi $. The arc length $l(r)$, measured from $\varphi =0$, is

22.18.4 | $$l(r)=(1/\sqrt{2})\mathrm{arccn}(r,1/\sqrt{2}).$$ | ||

Inversely:

22.18.5 | $$r=\mathrm{cn}(\sqrt{2}l,1/\sqrt{2}),$$ | ||

and

22.18.6 | $x$ | $=\mathrm{cn}(\sqrt{2}l,1/\sqrt{2})\mathrm{dn}(\sqrt{2}l,1/\sqrt{2}),$ | ||

$y$ | $=\mathrm{cn}(\sqrt{2}l,1/\sqrt{2})\mathrm{sn}(\sqrt{2}l,1/\sqrt{2})/\sqrt{2}.$ | |||

With $k\in [0,1]$ the mapping $z\to w=\mathrm{sn}(z,k)$ gives a conformal map of the closed rectangle $[-K,K]\times [0,{K}^{\prime}]$ onto the half-plane $\mathrm{\Im}w\ge 0$, with $0,\pm K,\pm K+\mathrm{i}{K}^{\prime},\mathrm{i}{K}^{\prime}$ mapping to $0,\pm 1,\pm {k}^{-2},\mathrm{\infty}$ respectively. The half-open rectangle $(-K,K)\times [-{K}^{\prime},{K}^{\prime}]$ maps onto $\mathrm{\u2102}$ cut along the intervals $(-\mathrm{\infty},-1]$ and $[1,\mathrm{\infty})$. See Akhiezer (1990, Chapter 8) and McKean and Moll (1999, Chapter 2) for discussions of the inverse mapping. Bowman (1953, Chapters V–VI) gives an overview of the use of Jacobian elliptic functions in conformal maps for engineering applications.

By use of the functions $\mathrm{sn}$ and $\mathrm{cn}$, parametrizations of algebraic equations, such as

22.18.7 | $$a{x}^{2}{y}^{2}+b({x}^{2}y+x{y}^{2})+c({x}^{2}+{y}^{2})+2dxy+e(x+y)+f=0,$$ | ||

in which $a,b,c,d,e,f$ are real constants, can be achieved in terms of
single-valued functions. This circumvents the cumbersome branch structure of
the multivalued functions $x(y)$ or $y(x)$, and constitutes the process of
*uniformization*; see Siegel (1988, Chapter II). See
Baxter (1982, p. 471) for an example from statistical mechanics.
Discussion of parametrization of the angles of spherical trigonometry in terms
of Jacobian elliptic functions is given in Greenhill (1959, p. 131)
and Lawden (1989, §4.4).

Algebraic curves of the form ${y}^{2}=P(x)$, where $P$ is a nonsingular
polynomial of degree 3 or 4 (see McKean and Moll (1999, §1.10)), are
*elliptic curves*, which are also considered in §23.20(ii). The special
case ${y}^{2}=(1-{x}^{2})(1-{k}^{2}{x}^{2})$ is in *Jacobian normal form*.
For any two points $({x}_{1},{y}_{1})$ and $({x}_{2},{y}_{2})$ on this curve, their *sum*
$({x}_{3},{y}_{3})$, always a third point on the curve, is defined by the Jacobi–Abel
addition law

22.18.8 | ${x}_{3}$ | $={\displaystyle \frac{{x}_{1}{y}_{2}+{x}_{2}{y}_{1}}{1-{k}^{2}{x}_{1}^{2}{x}_{2}^{2}}},$ | ||

${y}_{3}$ | $={\displaystyle \frac{{y}_{1}{y}_{2}+{x}_{2}(-(1+{k}^{2}){x}_{1}+2{k}^{2}{x}_{1}^{3})}{1-{k}^{2}{x}_{1}^{2}{x}_{2}^{2}}}+{x}_{3}{\displaystyle \frac{2{k}^{2}{x}_{1}{y}_{1}{x}_{2}^{2}}{1-{k}^{2}{x}_{1}^{2}{x}_{2}^{2}}},$ | |||

a construction due to Abel; see Whittaker and Watson (1927, pp. 442, 496–497). This provides an abelian group structure, and leads to important results in number theory, discussed in an elementary manner by Silverman and Tate (1992), and more fully by Koblitz (1993, Chapter 1, especially §1.7) and McKean and Moll (1999, Chapter 3). The existence of this group structure is connected to the Jacobian elliptic functions via the differential equation (22.13.1). With the identification $x=\mathrm{sn}(z,k)$, $y=d(\mathrm{sn}(z,k))/dz$, the addition law (22.18.8) is transformed into the addition theorem (22.8.1); see Akhiezer (1990, pp. 42, 45, 73–74) and McKean and Moll (1999, §§2.14, 2.16). The theory of elliptic functions brings together complex analysis, algebraic curves, number theory, and geometry: Lang (1987), Siegel (1988), and Serre (1973).