evenKernel_undef — Mathlib

English

If x ≤ 0, the even kernel is not defined by the series and is taken to be 0.

Русский

Если x ≤ 0, для четкого ядра série не существует, и оно принимается равным 0.

LaTeX

$$$$\\text{If } x \\le 0,\\quad \\operatorname{evenKernel}(a,x) = 0.$$$$

Lean4

/-- For `x ≤ 0` the defining sum diverges, so the kernel is 0. -/
theorem evenKernel_undef (a : UnitAddCircle) {x : ℝ} (hx : x ≤ 0) : evenKernel a x = 0 := by
  induction a using QuotientAddGroup.induction_on with
  | H a' => simp [← ofReal_inj, evenKernel_def, jacobiTheta₂_undef _ (by simpa : (I * ↑x).im ≤ 0)]

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