In mathematics, the half-period ratio τ of an elliptic function is the ratio
of the two half-periods and of the elliptic function, where the elliptic function is defined in such a way that
is in the upper half-plane.
Quite often in the literature, ω1 and ω2 are defined to be the periods of an elliptic function rather than its half-periods. Regardless of the choice of notation, the ratio ω2/ω1 of periods is identical to the ratio (ω2/2)/(ω1/2) of half-periods. Hence, the period ratio is the same as the "half-period ratio".
Note that the half-period ratio can be thought of as a simple number, namely, one of the parameters to elliptic functions, or it can be thought of as a function itself, because the half periods can be given in terms of the elliptic modulus or in terms of the nome.
See the pages on quarter period and elliptic integrals for additional definitions and relations on the arguments and parameters to elliptic functions.
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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.
In mathematics, the modular lambda function λ(τ) is a highly symmetric Holomorphic function on the complex upper half-plane. It is invariant under the fractional linear action of the congruence group Γ(2), and generates the function field of the corresponding quotient, i.e., it is a Hauptmodul for the modular curve X(2). Over any point τ, its value can be described as a cross ratio of the branch points of a ramified double cover of the projective line by the elliptic curve , where the map is defined as the quotient by the [−1] involution.
En mathématiques, une forme modulaire est une fonction analytique sur le demi-plan de Poincaré satisfaisant à une certaine sorte d'équation fonctionnelle et de condition de croissance. La théorie des formes modulaires est par conséquent dans la lignée de l'analyse complexe mais l'importance principale de la théorie tient dans ses connexions avec le théorème de modularité et la théorie des nombres.