jacobiTheta₂'_add_left — Mathlib

English

Quasi-periodicity for θ₂' in z with shift by τ: θ₂'(z+τ, τ) = exp(-π i (τ+2z)) [θ₂'(z, τ) − 2π i z θ₂(z, τ)].

Русский

Квазипериодичность θ₂' по z: θ₂'(z+τ, τ) = e^{-π i (τ+2z)} [θ₂'(z, τ) − 2π i z θ₂(z, τ)].

LaTeX

$$$\theta_2'(z+\tau, \tau) = e^{-\pi i (\tau+2z)} (\theta_2'(z, \tau) - 2\pi i z \theta_2(z, \tau))$$$

Lean4

theorem jacobiTheta₂'_add_left (z τ : ℂ) : jacobiTheta₂' (z + 1) τ = jacobiTheta₂' z τ :=
  by
  unfold jacobiTheta₂' jacobiTheta₂'_term jacobiTheta₂_term
  refine tsum_congr (fun n ↦ ?_)
  simp only [mul_add, Complex.exp_add, mul_one, mul_comm _ (n : ℂ), exp_int_mul_two_pi_mul_I, mul_one]

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