In
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
:
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
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, ω
1 and ω
2 are defined to be the periods of an elliptic function rather than its half-periods. Regardless of the choice of notation, the ratio ω
2/ω
1 of periods is identical to the ratio (ω
2/2)/(ω
1/2) 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.
See also
*
Modular form
In mathematics, a modular form is a (complex) analytic function on the upper half-plane satisfying a certain kind of functional equation with respect to the Group action (mathematics), group action of the modular group, and also satisfying a grow ...
*
Nome
References
* Milton Abramowitz and Irene A. Stegun, ''
Handbook of Mathematical Functions'', (1964) Dover Publications, New York. See chapters 16 and 17.
Elliptic functions
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