LemmaStatement: AI-adaptedProof: 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 composition of convergent complex power series has a convergent local power-series expansion when the inner sum maps the centre to the outer centre

Statement

Let $H (z) = \sum_{n \geq 1} h_{n} (z - a)^{n}$ converge near $a$ , and let $G (w) = \sum_{m \geq 0} g_{m} w^{m}$ converge for $∣ w ∣ < R$ for some $R > 0$ . If $H (a) = 0$ , then $G (H (z))$ has a convergent power-series expansion about $a$ on some neighbourhood of $a$ . The conclusion includes constant $H$ .

Facts & Assumptions

Given: Power series $H, G$ as in the Statement.

[L1]

Products of locally convergent complex power series are given by their Cauchy-product coefficients (Products of convergent complex power series are represented by their Cauchy-product coefficients on the common disc).

[L2]

A complex power series converges absolutely on every closed subdisc strictly inside its radius (A complex power series converges absolutely and uniformly on every closed subdisc strictly inside its disc of convergence).

[L3]

An absolutely convergent complex series may be rearranged without changing its sum (Every absolutely convergent complex series converges, and rearrangements preserve its sum).

Proof

technique · direct

1.1

Choose $ρ > 0$ inside the radius of $H$ . By [L2], $A = \sum_{n \geq 1} ∣ h_{n} ∣ ρ^{n}$ is finite. Since $R > 0$ , choose $0 < r \leq ρ$ and $0 \leq q < R$ with $(r / ρ) A \leq q$ ; then $\sum_{n \geq 1} ∣ h_{n} ∣ r^{n} \leq (r / ρ) A \leq q$ . If $A = 0$ , then $H \equiv 0$ and this estimate holds with $q = 0$ .

L2choosealgebra

2.1

Repeated use of [L1] expands each $H (z)^{m}$ as a power series about $a$ . Its absolute coefficient sum at radius $r$ is bounded by $q^{m}$ from step 1.1, and [L2] applied to $G$ gives convergence of the scalar majorant $\sum ∣ g_{m} ∣ q^{m}$ .

step 1.1L1L2

3.1

The resulting double series is absolutely convergent, so [L3] regroups it by powers of $z - a$ and produces the desired local series. If $H \equiv 0$ , it reduces to the constant $G (0)$ .

step 2.1L3∎

Depends on

Used by

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 52 results over 13 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

Power-series supplementary notes, Colby College (standard reference, not scraped)