Elliptic Modulus

The elliptic modulus is a quantity used in elliptic integrals and elliptic functions defined to be , where is the parameter. An elliptic integral is written when the parameter is used, whereas it is usually written where the elliptic modulus is used. The elliptic modulus tends to be more commonly used than the parameter (Abramowitz and Stegun 1972, p. 337; Whittaker and Watson 1990, p. 479), although most of Abramowitz and Stegun (1972, pp. 587-607), i.e., the entire chapter on elliptic integrals, and the Wolfram Language's EllipticE, EllipticF, EllipticK, EllipticPi, etc., use the parameter.

The elliptic modulus can be computed explicitly in terms of Jacobi theta functions of zero argument and with nome by

(1)

The real period and imaginary period are given by

(2)

(3)

where is a complete elliptic integral of the first kind and the complementary modulus is defined by

(4)

with the modulus.