CorollaryStatement: Literature-sourcedProof: AI-adaptedSession-authored (Fable 5 assisted)precheck passjudge pass (deepseek-v4-pro + gpt-5.6-terra)audited 2026-08-16

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A complex power-series representation about a fixed centre has unique coefficients

Statement

If two complex power series about the same centre $a$ represent the same function on a neighbourhood of $a$ , then their coefficients agree term by term.

Facts & Assumptions

Given: Representations $f (z) = \sum c_{n} (z - a)^{n} = \sum d_{n} (z - a)^{n}$ on one neighbourhood of $a$ .

[L1]

In any power-series representation about $a$ , the coefficient of order $n$ is $f^{(n)} (a) / n!$ (The coefficients of a complex power series are its derivatives at the centre divided by the corresponding factorials).

Proof

technique · direct

1.1

Both series represent the same function on a neighbourhood, so their derivatives of every order at $a$ are the same.

given

2.1

Applying [L1] to both representations gives $c_{n} = f^{(n)} (a) / n! = d_{n}$ for every $n$ , including $n = 0$ .

step 1.1L1∎

Depends on

The coefficients of a complex power series are its derivatives at the centre divided by the corresponding factorials

Used by

A convergent complex power series with nonzero constant term has a convergent reciprocal power series locally Lemma

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 23 results over 9 levels. An arrow runs from a result to what uses it, and this result sits at the bottom with a heavier outline. Click the chart to enlarge it.

Sources

L. Ahlfors, Complex Analysis, 3rd ed., Ch. 2 (standard reference, not scraped)