Half-period ratio

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In mathematics, the half-period ratio τ of an elliptic function j is the ratio

$\tau = \frac{\omega_2}{\omega_1}$

of the two half-periods ω₁ and ω₂ of j, where j is defined in such a way that

$\Im(\tau) > 0$

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.