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


x and y are generally considered variables and a and b 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 x and y, but if a and b are instead considered as variables, the resulting equation


is quartic in a and b.

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


where k is the elliptic modulus. An elliptic integral is written I(phi|m) when the parameter is used, whereas it is usually written I(phi,k) 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


where m is the parameter.

Let q be the nome, k the elliptic modulus, where m=k^2. Then


where K(m) is the complete elliptic integral of the first kind, and K^'(m)=K(m^'). Then the inverse of q(m) is given by


where theta_i(q) is a Jacobi theta function.

See also

Elliptic Characteristic, Elliptic Integral, Elliptic Integral of the First Kind, Elliptic Modulus, Half-Period Ratio, Jacobi Amplitude, Jacobi Theta Functions, Modular Angle, Nome, Parametric Equations, Parametric Curve, Parametric Surface, Quadric Parameter, Variable

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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.

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Cite this as:

Weisstein, Eric W. "Parameter." From MathWorld--A Wolfram Web Resource.

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