Nome

The nome associated with an elliptic modulus is defined by

			(1)
			(2)
			(3)

(Borwein and Borwein 1987, pp. 41, 109 and 114), where is the half-period ratio, is the complete elliptic integral of the first kind, and is the elliptic modulus. In terms of the parameter , write . The Wolfram Language function EllipticNomeQ[m] computes .

Extreme care is needed when consulting the literature, as it is common in the theory of modular functions (and in particular the Dedekind eta function) to use the symbol to denote , i.e., the square of the usual nome (e.g., Berndt 1993, p. 139). In this work, the modular version of is denoted

(4)

(e.g., Borwein and Borwein 1987, p. 118).

The nome is plotted above in the complex -plane.

For , 2, 3, and 4, notations for Jacobi theta functions involving the nome include

(5)

where is the half-period ratio (Whittaker and Watson 1990, p. 464) and

(6)

Special values, with endpoint values understood as limits from , include

			(7)
			(8)
			(9)

The nome as a function of the parameter has the Maclaurin series

(10)

(OEIS A002639 and A119349).

For , the nome has derivative

(11)

where uses the elliptic modulus convention of the defining formula above.

There exists a nonlinear third-order differential equation

(12)

for (Bertrand and Zudilin 2003; Trott 2006, pp. 29-31).

For , this equation has the compact form

(13)

where denotes the Schwarzian derivative. To obtain this form, write and use

(14)

The two complete elliptic integrals of the first kind are linearly independent solutions of the second-order ordinary differential equation

(15)

The local inverse function of is the elliptic lambda function .