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.

REFERENCES:

Abramowitz, M. and Stegun, I. A. (Eds.). Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, p. 590, 1972.

Borwein, J. M. and Borwein, P. B. Pi & the AGM: A Study in Analytic Number Theory and Computational Complexity. New York: Wiley, p. 35, 1987.

Tölke, F. "Parameterfunktionen." Ch. 3 in Praktische Funktionenlehre, zweiter Band: Theta-Funktionen und spezielle Weierstraßsche Funktionen. Berlin: Springer-Verlag, pp. 83-115, 1966.

Whittaker, E. T. and Watson, G. N. A Course in Modern Analysis, 4th ed. Cambridge, England: Cambridge University Press, 1990.