The term "parameter" is used in a number of ways in mathematics. In general, mathematical functions may have a number of arguments. Arguments that are typically varied when plotting, performing mathematical operations, etc., are termed "variables," while those that are not explicitly varied in situations of interest are termed "parameters." For example, in the standard equation of an ellipse

(1)

 and  are generally considered variables and  and  are considered parameters. The decision on which arguments to consider variables and which to consider parameters may be historical or may be based on the application under consideration. However, the nature of a mathematical function may change depending on which choice is made. For example, the above equation is quadratic in  and , but if  and  are instead considered as variables, the resulting equation

(2)

is quartic in  and .

In the theory of elliptic integrals, "the" parameter is denoted  and is defined to be

(3)

where  is the elliptic modulus. 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 complementary parameter is defined by

(4)

where  is the parameter.

Let  be the nome,  the elliptic modulus, where . Then

(5)

where  is the complete elliptic integral of the first kind, and . Then the inverse of  is given by

(6)

where  is a Jacobi theta function.

REFERENCES:

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

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