period ratio

τ is the ratio τ = ω_2/ω_1 of the two half-periods ω_1 and ω_2 of an elliptic function defined such that the imaginary part ℑ[τ]>0. The notation t is sometimes used instead of τ. The half-period ratio is most commonly encountered in the definition of the nome q as q congruent e^(π i τ) (Borwein and Borwein 1987, pp. 41, 109, and 114, p. 463) where K(k) is the complete elliptic integral of the first kind, m = k^2 is the parameter, k is the elliptic modulus, K'(k) = K(k'), and k' is the complementary elliptic modulus.

elliptic invariants | elliptic modulus | half-period | inverse nome | Jacobi theta functions | modular angle | nome | parameter | Weierstrass elliptic function