CorollaryStatement: Literature-sourcedProof: AI-adaptedSession-authored (Fable 5 assisted)precheck passaudited 2026-08-21

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A locally uniformly convergent series of holomorphic functions may be differentiated term by term

Statement

Let $Ω \subseteq C$ be open and let $g_{j} : Ω \to C$ be holomorphic. Suppose the sequence of partial sums of $\sum_{j \geq 0} g_{j}$ converges locally uniformly to $g$ . Then $g$ is holomorphic, and for every natural $k$ ,

$g^{(k)} = \sum_{j \geq 0} g_{j}^{(k)},$

where the derivative series converges locally uniformly.

Facts & Assumptions

Given: Holomorphic functions $g_{j}$ on a common open set $Ω$ and locally uniform convergence of their complex-series partial sums as defined in Complex series, absolute convergence, complex power series, and radius of convergence.

[L1]

Complex differentiation is linear, and every constant function has derivative zero (Linearity, product, reciprocal, and quotient rules for complex derivatives).

[L2]

A locally uniform limit of holomorphic functions is holomorphic, and for every natural $k$ the $k$ th derivatives converge locally uniformly to the $k$ th derivative of the limit (Locally uniform limits of holomorphic functions are holomorphic and their derivatives converge locally uniformly).

Proof

technique · direct

1.1

For the finite partial sum $S_{N} = \sum_{j < N} g_{j}$ , induction with [L1] gives $S_{N}^{(k)} = \sum_{j < N} g_{j}^{(k)}$ for every natural $k$ ; the empty partial sum is the zero holomorphic function.

L1algebra

2.1

Apply [L2] to the locally uniformly convergent sequence $(S_{N})$ : its limit $g$ is holomorphic and $S_{N}^{(k)} \to g^{(k)}$ locally uniformly for every natural $k$ .

step 1.1L2

3.1

By step 1.1, the sequence in step 2.1 is exactly the partial-sum sequence of $\sum_{j \geq 0} g_{j}^{(k)}$ , proving the displayed termwise derivative formula; at $k = 0$ it is the original series.

step 1.1step 2.1∎

Depends on

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Sources

Lars Ahlfors, Complex Analysis, 3rd ed., Ch. 5 §1.1 (standard reference, not scraped)
E. Stein and R. Shakarchi, Complex Analysis, Ch. 2 §5.2 (standard reference, not scraped)