Modular Transformations

Abstract

In sections 1.7 and 1.8, we studied the effect on the theta functions of transforming from a parameter τ to a parameter τ′, where τ′ = − 1/τ and τ′ = 2τ respectively. Then, in sections 2.6 and 3.9, it was shown that these transformations change the modulus of the Jacobian elliptic functions from k to $k' = \sqrt {(1 - {k^2})}$ and from k to k ₁ = (1− k′)/(1 + k′)respectively; the modulus transformation k = 1/k was also examined.