Half-period ratio
Encyclopedia
In mathematics
, the half-period ratio τ of an elliptic function
j is the ratio
of the two half-periods
ω1 and ω2 of j, where j is defined in such a way that
is in the upper half-plane.
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.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, the half-period ratio τ of an elliptic function
Elliptic function
In complex analysis, an elliptic function is a function defined on the complex plane that is periodic in two directions and at the same time is meromorphic...
j is the ratio
of the two half-periods
Fundamental pair of periods
In mathematics, a fundamental pair of periods is an ordered pair of complex numbers that define a lattice in the complex plane. This type of lattice is the underlying object with which elliptic functions and modular forms are defined....
ω1 and ω2 of j, where j is defined in such a way that
is in the upper half-plane.
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.