Nome (mathematics)

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. 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: 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.

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