TheoremStatement: Literature-sourcedProof: AI-adaptedSession-authored (Fable 5 assisted)precheck passjudge pass (deepseek-v4-pro + gpt-5.6-terra)audited 2026-08-26

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Locally uniform limits of holomorphic functions are holomorphic, with locally uniform convergence of all derivatives

Statement

Let $m \geq 1$ , let $U \subseteq C^{m}$ be open, let each $f_{n} : U \to C$ be holomorphic, and suppose $f_{n} \to f$ locally uniformly on $U$ : every point of $U$ has a neighbourhood on which the convergence is uniform. Then $f$ is holomorphic on $U$ , and for every multi-index $α$

$\partial_{z}^{α} f_{n} ⟶ \partial_{z}^{α} f$

locally uniformly on $U$ .

Facts & Assumptions

Given: An open $U \subseteq C^{m}$ , holomorphic $f_{n} : U \to C$ and $f : U \to C$ with $f_{n} \to f$ locally uniformly in the sense of Locally uniform convergence on an open subset of the complex plane is compact convergence and Uniform convergence and the uniformly Cauchy condition for complex-valued functions, with the componentwise dictionary; $C^{m}$ is read through Complex $m$ -space and its real coordinate dictionary.

[L1]

A continuous separately holomorphic function on an open subset of $C^{m}$ is holomorphic (Osgood's lemma: continuous and separately holomorphic implies holomorphic).

[L2]

For $g$ holomorphic on $Δ_{ρ} (a)$ and a polyradius $r$ with $r_{k} < ρ_{k}$ , $∣ \partial_{z}^{α} g (a) ∣ \leq α! \sup_{Γ_{r} (a)} ∣ g ∣ \prod_{k < m} r_{k}^{- α_{k}}$ (Cauchy estimates for mixed derivatives on a polydisc).

[L3]

If holomorphic functions of one variable converge locally uniformly on an open subset of $C$ , the limit is holomorphic (Locally uniform limits of holomorphic functions are holomorphic and their derivatives converge locally uniformly).

[L4]

A uniform limit of continuous complex-valued functions on a metric space is continuous (A uniform limit of continuous complex-valued functions is continuous).

[L5]

Separate holomorphy is holomorphy of each slice on its open slice domain (Separately holomorphic functions).

[L6]

A holomorphic function of several variables is continuous and separately holomorphic (A holomorphic function of several variables is continuous and separately holomorphic); every iterated complex partial derivative of a holomorphic function is holomorphic (Holomorphic functions of several variables are smooth and their complex derivatives are holomorphic); differences of holomorphic functions are holomorphic (Sums, products and nonvanishing quotients of holomorphic functions are holomorphic).

[L7]

$Δ_{r} (a)$ , ${\overline{Δ}}_{r} (a)$ and $Γ_{r} (a)$ are defined coordinatewise (Balls, polydiscs and the distinguished boundary in $C^{m}$ ); multi-index notation is that of $C^{k}$ maps and multi-index derivative notation in Euclidean space.

[L8]

A set is open exactly when each of its points admits a ball inside it (The metric topology: a set is open when every one of its points has a ball around it inside the set; closed means open complement, Open ball, closed ball and sphere in a metric space).

[L9]

$∣ z + w ∣ \leq ∣ z ∣ + ∣ w ∣$ for complex $z, w$ (Conjugation is an involutive real-field automorphism, $z \overline{z} = ∣ z ∣^{2}$ , and modulus is definite, multiplicative, and subadditive).

Proof

technique · direct

1.1

Each $f_{n}$ is continuous by [L6], and locally uniform convergence makes $f$ continuous at every point by [L4] applied on a neighbourhood where the convergence is uniform.

givenL4L6

1.2

Fix $a \in U$ and $k < m$ , and let $V$ be the $k$ th slice domain of $U$ through $a$ , an open subset of $C$ by [L5]. The slices of the $f_{n}$ are holomorphic on $V$ by [L6], and they converge to the slice of $f$ locally uniformly on $V$ , since a neighbourhood in $U$ of a point of the slice meets the slice in a neighbourhood there. So [L3] makes the slice of $f$ holomorphic on $V$ , and $f$ is separately holomorphic by [L5].

givenL3L5L6L8

2.1

By steps 1.1 and 1.2 the limit $f$ is continuous and separately holomorphic on $U$ , so [L1] makes it holomorphic.

step 1.1step 1.2L1

3.1

Fix $a \in U$ and a multi-index $α$ . By [L8] choose $ε > 0$ with the ball $B (a, ε) \subseteq U$ and put $ρ_{k} = ε / (2 \sqrt{m})$ , so $Δ_{ρ} (a) \subseteq U$ by [L7] and [L9], and put $r_{k} = ρ_{k} / 2$ . Shrinking $ε$ if necessary, the convergence $f_{n} \to f$ is uniform on ${\overline{Δ}}_{ρ} (a)$ .

step 2.1L7L8L9

4.1

Fix $b \in Δ_{r} (a)$ and put $σ_{k} : = ρ_{k} - ∣ b_{k} - a_{k} ∣$ for each $k < m$ . Then $σ_{k} > r_{k}$ because $∣ b_{k} - a_{k} ∣ < r_{k} = ρ_{k} / 2$ , and if $∣ ζ_{k} - b_{k} ∣ < σ_{k}$ then $∣ ζ_{k} - a_{k} ∣ \leq ∣ ζ_{k} - b_{k} ∣ + ∣ b_{k} - a_{k} ∣ < ρ_{k}$ , so $Δ_{σ} (b) \subseteq Δ_{ρ} (a) \subseteq U$ by [L7] and [L9]. Also $Γ_{r} (b) \subseteq {\overline{Δ}}_{ρ} (a)$ because $∣ b_{k} - a_{k} ∣ < r_{k}$ and $r_{k} + r_{k} = ρ_{k}$ . The difference $f_{n} - f$ is holomorphic on $U$ by [L6] and step 2.1, so [L2] applied on $Δ_{σ} (b)$ with inner polyradius $r$ gives $∣ \partial_{z}^{α} f_{n} (b) - \partial_{z}^{α} f (b) ∣ \leq α! (\sup_{{\overline{Δ}}_{ρ} (a)} ∣ f_{n} - f ∣) \prod_{k < m} r_{k}^{- α_{k}}$ .

step 3.1L2L6L7L9

5.1

The right-hand side of step 4.1 does not depend on $b$ and tends to $0$ by the uniform convergence of step 3.1, so $\partial_{z}^{α} f_{n} \to \partial_{z}^{α} f$ uniformly on the neighbourhood $Δ_{r} (a)$ of $a$ . Since $a \in U$ and $α$ were arbitrary, the convergence is locally uniform for every multi-index.

step 3.1step 4.1L6L7∎

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Sources

J. Lebl, Tasty Bits of Several Complex Variables, §1.2 (standard reference, not scraped)
H. P. Boas, Lecture Notes on Multidimensional Complex Analysis, Ch. 2 (standard reference, not scraped)
M. Jabbari, Notes for Analysis and Geometry of Several Complex Variables, §3.1 (standard reference, not scraped)