Mathematics · Complex Analysis
Let h be a holomorphic function on ℂ ∖ {0} such that lim|z| → &infty; h(z) = 0. For every n ≥ 1, consider fn(z) = ∑k=1n h(zk). Let 𝔻 denote the open unit disc { z ∈ ℂ : |z| < 1 }. Which of the following statements is necessarily true?
Verified Solution4 OptionsUpdated: October 2026👁️1 View
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- 1The sequence {fn}n ≥ 1 converges pointwise to a holomorphic function on { z ∈ ℂ : |z| > 1 }.✓
- 2The sequence {fn}n ≥ 1 converges pointwise to a holomorphic function on 𝔻 ∖ [0, 1).
- 3For all z ∈ 𝔻 ∖ {0}, the sequence {fn(z)}n ≥ 1 converges.
- 4For all z with |z| ≥ 1, the sequence {fn(z)}n ≥ 1 converges.
Correct answer
The sequence {fn}n ≥ 1 converges pointwise to a holomorphic function on { z ∈ ℂ : |z| > 1 }.
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