Θ₂ term imag axis re
Θ₂_term_imag_axis_re
Plain-language statement
Each term Θ₂_term n (I*t) has positive real part equal to exp(-π(n+1/2)²t) for t > 0.
Exact Lean statement
lemma Θ₂_term_imag_axis_re (n : ℤ) (t : ℝ) (ht : 0 < t) :
(Θ₂_term n ⟨I * t, by simp [ht]⟩).re =
Real.exp (-Real.pi * ((n : ℝ) + 1/2) ^ 2 * t)Formal artifact
Lean source
lemma Θ₂_term_imag_axis_re (n : ℤ) (t : ℝ) (ht : 0 < t) : (Θ₂_term n ⟨I * t, by simp [ht]⟩).re = Real.exp (-Real.pi * ((n : ℝ) + 1/2) ^ 2 * t) := by unfold Θ₂_term change (cexp (Real.pi * I * ((n : ℂ) + 1 / 2) ^ 2 * (I * t))).re = _ have hexpr : Real.pi * I * ((n : ℂ) + 1 / 2) ^ 2 * (I * ↑t) = (-(Real.pi * ((n : ℝ) + 1/2) ^ 2 * t) : ℝ) := by have hI : I ^ 2 = -1 := I_sq push_cast ring_nf simp only [hI] ring rw [hexpr] rw [Complex.exp_ofReal_re] ring_nf- Project
- Sphere Packing in Dimension 8
- License
- Apache-2.0
- Commit
- acfc6204e65a
- Source
- SpherePacking/ModularForms/JacobiTheta/Basic.lean:694-708
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