Nome (mathematics)

In mathematics, specifically the theory of elliptic functions, the nome is a special function and is given by

${\displaystyle q=e^{-{\frac {\pi K'}{K}}}=e^{\frac {{\rm {i}}\pi \omega _{2}}{\omega _{1}}}=e^{{\rm {i}}\pi \tau }\,}$

where K and iK′ are the quarter periods, and ω₁ and ω₂ are the fundamental pair of periods, and τ = iK ′/K = ω₂/ω₁ is the half-period ratio. 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. That is, the mappings between these various symbols are both 1-to-1 and onto, and so can be inverted: the quarter periods, the half-periods and the half-period ratio can be explicitly written as functions of the nome. Explicit expressions for the quarter periods, in terms of the nome, are given in the linked article. Conversely, the above can be taken as an explicit expression for the nome, in terms of the other quantities.

Thus, the nome can be take to be either a function, or as a parameter; conversely, the quarter and half periods can be taken either as functions, or as parameters; specifying any one is sufficient to uniquely determine all of the others; they are all functions of one-another.

Notationally, the quarter periods K and iK′ are usually used only in the context of the Jacobian elliptic functions, whereas the half-periods ω₁ and ω₂ are usually used only in the context of Weierstrass elliptic functions. Some authors, notably Apostol, use ω₁ and ω₂ to denote whole periods rather than half-periods.

The nome is frequently used as a value with which elliptic functions and modular forms can be described; on the other hand, it can also be thought of as function, because the quarter periods are functions of the elliptic modulus. This ambiguity occurs because for real values of the elliptic modulus, the quarter periods and thus the nome are uniquely determined.

The complementary nome q₁ is given by

${\displaystyle q_{1}=e^{-{\frac {\pi K}{K'}}}.\,}$

Some sources however use convention ${\displaystyle q=e^{{2{\rm {i}}}\pi \tau }}$ or ${\displaystyle q=e^{{2{\rm {i}}}\pi z}}$.

See the articles on quarter period and elliptic integrals for additional definitions and relations on the nome.