In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, specifically the theory of
elliptic functions
In the mathematical field of complex analysis, elliptic functions are a special kind of meromorphic functions, that satisfy two periodicity conditions. They are named elliptic functions because they come from elliptic integrals. Originally those i ...
, the nome is a
special function
Special functions are particular mathematical functions that have more or less established names and notations due to their importance in mathematical analysis, functional analysis, geometry, physics, or other applications.
The term is defined b ...
that belongs to the non-elementary functions. This function is of great importance in the description of the elliptic functions, especially in the description of the modular identity of the
Jacobi theta function
In mathematics, theta functions are special functions of several complex variables. They show up in many topics, including Abelian varieties, moduli spaces, quadratic forms, and solitons. As Grassmann algebras, they appear in quantum field theo ...
, the
Hermite elliptic transcendents and the
Weber modular functions, that are used for solving equations of higher degrees.
Definition
The nome function is given by
:
where ''
'' and
are the
quarter period In mathematics, the quarter periods ''K''(''m'') and i''K'' ′(''m'') are special functions that appear in the theory of elliptic functions.
The quarter periods ''K'' and i''K'' ′ are given by
:K(m)=\int_0^ \frac
and
:K'(m) ...
s, and
and
are the
fundamental pair of periods In mathematics, a fundamental pair of periods is an ordered pair of complex numbers that define a lattice in the complex plane. This type of lattice is the underlying object with which elliptic functions and modular forms are defined.
Definition ...
, and
is the
half-period ratio
In mathematics, the half-period ratio τ of an elliptic function is the ratio
:\tau = \frac
of the two half-periods \frac and \frac of the elliptic function, where the elliptic function is defined in such a way that
:\Im(\tau) > 0
is in the ...
. The nome can be taken to be a function of any one of these quantities; conversely, any one of these quantities can be taken as functions of the nome. Each of them uniquely determines the others when