Jacobi's theta function #
This file defines the one-variable Jacobi theta function
$$\theta(\tau) = \sum_{n \in \mathbb{Z}} \exp (i \pi n ^ 2 \tau),$$
and proves the modular transformation properties θ (τ + 2) = θ τ
and
θ (-1 / τ) = (-I * τ) ^ (1 / 2) * θ τ
, using Poisson's summation formula for the latter. We also
show that θ
is differentiable on ℍ
, and θ(τ) - 1
has exponential decay as im τ → ∞
.
Jacobi's one-variable theta function ∑' (n : ℤ), exp (π * I * n ^ 2 * τ)
.
Instances For
The norm of jacobiTheta τ - 1
decays exponentially as im τ → ∞
.