Half-period ratio

In mathematics, the half-period ratio τ of an elliptic function (such as Klein's j-invariant) is the ratio In mathematics, the half-period ratio τ of an elliptic function (such as Klein's j-invariant) is the ratio of the two half-periods ω 1 2 {displaystyle {frac {omega _{1}}{2}}} and ω 2 2 {displaystyle {frac {omega _{2}}{2}}} of j, where j 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. This follows because Klein's j-invariant is surjective onto the complex plane; it gives a bijection between isomorphism classes of elliptic curves and the complex numbers. See the pages on quarter period and elliptic integrals for additional definitions and relations on the arguments and parameters to elliptic functions.