DefinitionDefinition: Literature-sourcedProof: Not applicablejudge pass (deepseek-v4-pro + gpt-5.6-terra)audited 2026-08-21

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

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AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

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The Taylor series of a holomorphic function at a point

Definition

Let $f$ be holomorphic on an open set $Ω \subseteq C$ , and let $a \in Ω$ . The Taylor series of $f$ at $a$ is $\sum_{n \geq 0} f^{(n)} (a) (z - a)^{n} / n!$ .

Every derivative $f^{(n)} (a)$ exists by All higher complex derivatives exist and satisfy Cauchy's integral formula on an interior circle, and $n!$ is the positive factorial of The factorial $n!$ and the falling factorial $n^{\underline{k}}$ , defined by recursion in $N$ , so each coefficient

$c_{n} : = \frac{f^{(n)} (a)}{n!}$

is a well-defined complex number. The resulting complex power series is understood according to Complex series, absolute convergence, complex power series, and radius of convergence. This definition names the formal series; its convergence and equality with $f$ are conclusions of the Taylor expansion theorem.

The Taylor series of a holomorphic function at a point

Definition

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