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In mathematics, the half-period ratio τ of an elliptic function 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 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 / of half-periods. Hence the period ratio is the same as the "half-period ratio".