tendsto_partialSum_prod — Mathlib

English

If f is analytic within s at x and s ∈ 𝒩𝑕𝑑Within t, then f is analytic within t at x.

Русский

Если f аналитична внутри s в x и s принадлежит nhdsWithin t, тогда f аналитична внутри t в x.

LaTeX

$$$\text{AnalyticWithinAt}_{\mathcal{K}}(f,s,x) \land s \in \mathcal{N}_{\!t}(x) \Rightarrow \text{AnalyticWithinAt}_{\mathcal{K}}(f,t,x)$$$

Lean4

/-- If a function admits a power series on a ball, then the partial sums
`p.partialSum n z` converges to `f (x + y)` as `n → ∞` and `z → y`. -/
theorem tendsto_partialSum_prod {y : E} (hf : HasFPowerSeriesOnBall f p x r) (hy : y ∈ EMetric.ball (0 : E) r) :
    Tendsto (fun (z : ℕ × E) ↦ p.partialSum z.1 z.2) (atTop ×ˢ 𝓝 y) (𝓝 (f (x + y))) :=
  (hf.hasFPowerSeriesWithinOnBall (s := univ)).tendsto_partialSum_prod hy (by simp)

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