False statementConstruction: AI-adaptedVerification: AI-adaptedSession-authored (Fable 5 assisted)precheck passjudge pass (deepseek-v4-pro + gpt-5.6-terra)audited 2026-08-21

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FALSE: every entire function with an antiderivative is a polynomial

Statement

False claim: Every entire function that has an entire antiderivative is a polynomial.

Facts & Assumptions

Given: The complex exponential function.

[L1]

The complex exponential is entire and satisfies $\exp^{'} = \exp$ on $C$ (The complex exponential is entire and its complex derivative is itself).

[L2]

For all complex $z, w$ , $\exp (z + w) = \exp z \exp w$ , and the complex exponential agrees with the real exponential on the real axis ( $\exp (z + w) = \exp z \exp w$ , and the complex exponential extends the real exponential).

[L3]

Every nonconstant complex polynomial has a complex root (Fundamental theorem of algebra: every nonconstant complex polynomial has a complex root).

[L4]

The normalized real exponential satisfies $\exp 0 = 1$ and has derivative $1$ at $0$ (Regular normalized multiplicative Cauchy equations characterize the exponential).

Refutation

technique · direct

1.1

By [L1], the complex exponential is entire and is its own entire antiderivative.

1.2

By [L2] and [L4], $\exp z \exp (- z) = \exp 0 = 1$ , so the exponential never vanishes; [L1] then makes its derivative nonzero everywhere, and it is nonconstant.

L1L2L4algebra

2.1

If the complex exponential were a nonconstant polynomial, [L3] would give it a complex zero, contradicting step 1.2.

step 1.2L3

3.1

It is not a constant polynomial because its derivative is nonzero by step 1.2, and step 2.1 excludes every nonconstant polynomial; together with step 1.1, the exponential is an entire nonpolynomial function with an entire antiderivative, refuting the claim.

step 1.1step 1.2step 2.1∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

28 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Lars Ahlfors, Complex Analysis, 3rd ed., Ch. 4 §2.3 (standard reference, not scraped)
E. Stein and R. Shakarchi, Complex Analysis, Ch. 2 §4 (standard reference, not scraped)
Steven G. Krantz, A Guide to Complex Variables, §3.1.4 (standard reference, not scraped)