TheoremStatement: AI-adaptedProof: AI-adaptedSession-authored (Fable 5 assisted)precheck passaudited 2026-07-31

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Real-analytic functions are closed under sums, products and compositions, and under quotients where the denominator is nonzero

Statement

On open real domains, sums and products of real-analytic functions are real analytic. If $g$ is nonzero throughout the domain, then $f/g$ is real analytic. If $g:U\to V$ and $f:V\to\mathbb R$ are real analytic, then $f\circ g$ is real analytic on $U$ .

Facts & Assumptions

Given: Real-analytic functions with compatible domains as in the statement.

[L1]

Each function has a convergent local power-series representation at every point (A real-analytic function on an open subset of $\mathbb{R}$ is locally represented by a convergent real power series).

[L2]

Products are represented by Cauchy products, local reciprocals exist when the constant term is nonzero, and local compositions have convergent expansions (Inside the common radius the product of two power-series sums is represented by the Cauchy product of their coefficients, A convergent real power series with nonzero constant term has a convergent reciprocal power series on a smaller neighbourhood, A composition of convergent real power series has a convergent power-series expansion wherever the inner series maps a neighbourhood into the outer disk of convergence).

Proof

technique · cases

1.1

For sums and products, fix a point and choose local series for the functions by [L1]. Termwise addition gives a convergent series for the sum, while [L2] gives one for the product.

assume-case algebraL1L2

1.2

For a quotient, the denominator is nonzero at the point, so its local series has nonzero constant term. By [L2] it has a local reciprocal series, whose Cauchy product with the numerator series represents the quotient.

assume-case quotientL1L2

1.3

For a composition, centre the outer series at the inner value. The inner series then has zero constant term, and absolute convergence lets one shrink the radius until the sum of the absolute values of its nonconstant terms is smaller than the outer radius. The composition lemma in [L2] then supplies a local series.

assume-case compositionL1L2

2.1

The cases in steps 1.1--1.3 are precisely the operations in the statement. Each construction works at every point of the relevant open domain, so the definition [L1] proves all assertions.

step 1.1step 1.2step 1.3L1cases-exhaustive∎

Depends on

Used by

A rational function with nonvanishing denominator is locally represented by geometric-series expansions Example

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 51 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

Analytic function, Encyclopedia of Mathematics (standard reference, not scraped)
Power series, Encyclopedia of Mathematics (standard reference, not scraped)
E. Randles, Supplementary Notes for Real Analysis (standard reference, not scraped)
Northwestern Math 320-2 lecture notes (standard reference, not scraped)