cosKernel_undef — Mathlib

English

cosKernel is identically zero for x ≤ 0, and negative a does not affect its zero property.

Русский

cosKernel тождественно равен нулю при x ≤ 0; ноль не зависит от знака a.

LaTeX

$$$$\\forall a, x:\\; x\\le 0 \\Rightarrow \\cosKernel(a,x) = 0.$$$$

Lean4

theorem cosKernel_undef (a : UnitAddCircle) {x : ℝ} (hx : x ≤ 0) : cosKernel a x = 0 := by
  induction a using QuotientAddGroup.induction_on with
  | H => simp [← ofReal_inj, cosKernel_def, jacobiTheta₂_undef _ (by simpa : (I * ↑x).im ≤ 0)]

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