The elliptic parameter 
is the square of the elliptic
modulus 
used in elliptic integrals and elliptic
functions,
 |
(1)
|
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).
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
 |
(2)
|
where 
is the elliptic parameter. The prime denotes complementation, not a derivative,
and 
, where 
is the complementary
modulus.
For 
, the nome is
 |
(3)
|
where 
denotes the complete elliptic
integral of the first kind in the elliptic parameter convention and 
. The inverse
function of 
is given by
 |
(4)
|
where 
is a Jacobi theta function (National Institute
of Standards and Technology).
