mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...

, the half-period ratio τ of an

elliptic function In the mathematical field of complex analysis, elliptic functions are a special kind of meromorphic functions, that satisfy two periodicity conditions. They are named elliptic functions because they come from elliptic integrals. Originally those in ...

is the ratio :

\tau = \frac

of the two half-periods

\frac

and

\frac

of the elliptic function, where the elliptic function is defined in such a way that :

\Im(\tau) > 0

is in the

upper half-plane In mathematics, the upper half-plane, \,\mathcal\,, is the set of points in the Cartesian plane with > 0. Complex plane Mathematicians sometimes identify the Cartesian plane with the complex plane, and then the upper half-plane corresponds to t ...

. 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 In mathematics, the modular lambda function λ(τ)\lambda(\tau) is not a modular function (per the Wikipedia definition), but every modular function is a rational function in \lambda(\tau). Some authors use a non-equivalent definition of "modular f ...

or in terms of the nome. See the pages on

quarter period In mathematics, the quarter periods ''K''(''m'') and i''K'' ′(''m'') are special functions that appear in the theory of elliptic functions. The quarter periods ''K'' and i''K'' ′ are given by :K(m)=\int_0^ \frac and :K'(m) ...

and

elliptic integrals In integral calculus, an elliptic integral is one of a number of related functions defined as the value of certain integrals, which were first studied by Giulio Fagnano and Leonhard Euler (). Their name originates from their originally arising i ...

for additional definitions and relations on the arguments and parameters to elliptic functions.

References

* Milton Abramowitz and Irene A. Stegun, '' Handbook of Mathematical Functions'', (1964) Dover Publications, New York. See chapters 16 and 17. Elliptic functions {{numtheory-stub

See also

References