In mathematics, specifically the theory of elliptic functions, the nome is a special function 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, 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 periods, and and are the fundamental pair of periods, and 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 . That is, when , 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 with , is not a single-valued function of . Explicit expressions for the quarter periods, in terms of the nome, are given in the linked article.
Notationally, the quarter periods and are usually used only in the context of the Jacobian elliptic functions, whereas the half-periods and are usually used only in the context of Weierstrass elliptic functions. Some authors, notably Apostol, use and 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 : .
The complementary nome is given by
Sometimes the notation is used for the square of the nome.
The mentioned functions and are called complete elliptic integrals of the first kind. They are defined as follows:
Applications
The nome solves the following equation:
This analogon is valid for the Pythagorean complementary modulus:
where are the complete Jacobi theta functions and is the complete elliptic integral of the first kind with modulus shown in the formula above. For the complete theta functions these definitions introduced by Sir Edmund Taylor Whittaker and George Neville Watson are valid:
These three definition formulas are written down in the fourth edition of the book A Course in Modern Analysis written by Whittaker and Watson on the pages 469 and 470. 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 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 is not part of the disk (or rather, corresponds to ). 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 (or of , or of and 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 . 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 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.
Curve sketching
Every real value of the interval is assigned to a real number between inclusive zero and inclusive one in the nome function . The elliptic nome function is axial symmetric to the ordinate axis. Thus: . The functional curve of the nome passes through the origin of coordinates with the slope zero and curvature plus one eighth. For the real valued interval the nome function is strictly left-curved.
Derivatives
The Legendre's relation is defined that way: