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.
