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22 Jacobian Elliptic FunctionsApplications

§22.18 Mathematical Applications

Contents

§22.18(i) Lengths and Parametrization of Plane Curves

Ellipse

22.18.1\left(x^{2}/a^{2}\right)+\left(y^{2}/b^{2}\right)=1,
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with a\geq b>0, is parametrized by

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

where \mathop{\mathcal{E}\/}\nolimits\!\left(u,k\right) is Jacobi’s epsilon function (§22.16(ii)).

Lemniscate

In polar coordinates, x=r\mathop{\cos\/}\nolimits\phi, y=r\mathop{\sin\/}\nolimits\phi, the lemniscate is given by r^{2}=\mathop{\cos\/}\nolimits\!\left(2\phi\right), 0\leq\phi\leq 2\pi. The arc length l(r), measured from \phi=0, is

Inversely:

and

For these and other examples see Lawden (1989, Chapter 4), Whittaker and Watson (1927, §22.8), and Siegel (1988, pp. 1–7).

§22.18(ii) Conformal Mapping

With k\in[0,1] the mapping z\to w=\mathop{\mathrm{sn}\/}\nolimits\left(z,k\right) gives a conformal map of the closed rectangle [-K,K]\times[0,K^{{\prime}}] onto the half-plane \imagpart{w}\geq 0, with 0,\pm K,\pm K+iK^{{\prime}},iK^{{\prime}} mapping to 0,\pm 1,\pm k^{{-2}},\infty respectively. The half-open rectangle (-K,K)\times[-K^{{\prime}},K^{{\prime}}] maps onto \Complex cut along the intervals (-\infty,-1] and [1,\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.

§22.18(iii) Uniformization and Other Parametrizations

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

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).

§22.18(iv) Elliptic Curves and the Jacobi–Abel Addition Theorem

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}=\frac{x_{1}y_{2}+x_{2}y_{1}}{1-k^{2}x_{1}^{2}x_{2}^{2}},
y_{3}=\frac{y_{1}y_{2}+x_{2}(-(1+k^{2})x_{1}+2k^{2}x_{1}^{3})}{1-k^{2}x_{1}^{2% }x_{2}^{2}}+x_{3}\frac{2k^{2}x_{1}y_{1}x_{2}^{2}}{1-k^{2}x_{1}^{2}x_{2}^{2}},
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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=\mathop{\mathrm{sn}\/}\nolimits\left(z,k\right), y=\ifrac{d(\mathop{\mathrm{sn}\/}\nolimits\left(z,k\right))}{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).

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