DefinitionDefinition: Literature-sourcedProof: Not applicable

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

Complex differentiability at a point, the complex derivative, holomorphic functions, and entire functions

Definition

Let $U \subseteq C$ be open, let $a \in U$ , and let $f : U \to C$ . The function $f$ is complex differentiable at $a$ if the limit

$\lim_{\begin{array}{c} h \to 0 \\ h \neq 0, a + h \in U \end{array}} \frac{f (a + h) - f (a)}{h}$

exists in the metric of The Euclidean metric, convergence, Cauchy sequences, and continuity on the complex plane. Its value is the complex derivative of $f$ at $a$ , denoted $f^{'} (a)$ . The increments are nonzero and remain in the domain; because $U$ is open, all sufficiently small increments are allowed.

The function $f$ is holomorphic on $U$ when it is complex differentiable at every point of $U$ . A function holomorphic on all of $C$ is entire. The word analytic is reserved for the local power-series notion.

Complex differentiability at a point, the complex derivative, holomorphic functions, and entire functions

Definition

Depends on

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