Nome (mathematics)
In mathematics, specifically the theory of elliptic functions, the nome is a special function and is given by where "K" and i"K"′ are the quarter periods, and ω and ω are the fundamental pair of periods, and τ = i"K" ′/"K" = ω/ω 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 when formula_2. That is, when formula_2, 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. For general formula_4 with formula_5, formula_6 is not a single-valued function of formula_7. Explicit expressions for the quarter periods, in terms of the nome, are given in the linked article. Notationally, the quarter periods "K" and i"K"′ are usually used only in the context of the Jacobian elliptic functions, whereas the half-periods ω and ω are usually used only in the context of Weierstrass elliptic functions. Some authors , notably Apostol, use ω and ω 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 formula_8: formula_9. The complementary nome "q" is given by Sometimes the notation formula_11 is used for the "square" of the nome. Applications. The nome solves the following equation: where formula_13 are the Jacobi theta functions and formula_14 is the complete elliptic integral of the first kind  with modulus formula_8. The nome is commonly used as the starting point for the construction of Lambert series, the q-series and more generally the q-analogs. That is, the half-period ratio τ is commonly used as a coordinate on the complex upper half-plane, typically endowed with the Poincaré metric to obtain the Poincaré half-plane model. The nome then serves as a coordinate on a punctured disk of unit radius; it is punctured because "q"=0 is not part of the disk (or rather, "q"=0 corresponds to τ → ∞). This endows the punctured disk with the Poincaré  metric. The upper half-plane (and the Poincaré disk, and the punctured disk) can thus be tiled with the fundamental domain, which is the region of values of the half-period ratio τ (or of "q", or of "K" and i"K"′ etc.) that uniquely determine a tiling of the plane by parallelograms. The tiling is referred to as the modular symmetry given by the modular group. Some functions that are periodic on the upper half-plane are called to as modular functions; the nome, the half-periods, the quarter-periods or the half-period ratio all provide different parameterizations for  these periodic functions. The prototypical modular function is Klein's j-invariant. It can be written as a function of either the half-period ratio τ or as a function of the nome "q". The series expansion in terms of the nome or the square of the nome (the "q"-expansion) is famously connected to the Fisher-Griess monster by means of monstrous moonshine. Euler's function arises as the prototype for "q"-series in general. The nome, as the "q" of "q"-series then arises in the theory of affine Lie algebras, essentially because (to put it poetically, but not factually) those  algebras describe the symmetries and isometries of Riemann surfaces.