with , is parametrized by
where is the eccentricity, and . The arc length in the first quadrant, measured from , is
where is Jacobi’s epsilon function (§22.16(ii)).
With the mapping gives a conformal map of the closed rectangle onto the half-plane , with mapping to respectively. The half-open rectangle maps onto cut along the intervals and . 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 and , parametrizations of algebraic equations, such as
in which are real constants, can be achieved in terms of single-valued functions. This circumvents the cumbersome branch structure of the multivalued functions or , 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 , where 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 is in Jacobian normal form. For any two points and on this curve, their sum , always a third point on the curve, is defined by the Jacobi–Abel addition law
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 , , 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).