CorollaryStatement: AI-generatedProof: AI-generatedSession-authored (Fable 5 assisted)precheck passjudge pass (deepseek-v4-pro + gpt-5.6-terra)audited 2026-08-21

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Every nonconstant entire function has dense image in the complex plane

Statement

Every nonconstant entire function has dense image in the complex plane.

Equivalently, if $f : C \to C$ is entire and nonconstant, then every nonempty open disc meets $f [C]$ .

Facts & Assumptions

Given: A nonconstant entire function $f$ and the usual metric topology on $C$ from The Euclidean metric, convergence, Cauchy sequences, and continuity on the complex plane.

[L1]

A subset of a topological space is dense exactly when it meets every nonempty open set, equivalently every nonempty member of a chosen basis (Dense, nowhere dense and codense subsets of a topological space, and the criterion by basic open sets).

[L2]

If a complex differentiable function is nonzero at a point, its reciprocal is complex differentiable there; linear combinations of complex differentiable functions are complex differentiable (Linearity, product, reciprocal, and quotient rules for complex derivatives).

[L3]

Every bounded entire function is constant (Liouville's theorem: every bounded entire function is constant).

Proof

technique · direct

1.1

Suppose the image of $f$ is not dense. By [L1], some nonempty open set misses it; choosing a point $w$ of that set and a metric ball contained in it gives $ε > 0$ with $D (w, ε) \cap f [C] = \emptyset$ .

givenL1choose

2.1

The function $f - w$ never vanishes, so [L2] makes $g : = 1 / (f - w)$ entire, and step 1.1 gives $∣ f (z) - w ∣ \geq ε$ and hence $∣ g (z) ∣ \leq 1 / ε$ for every $z$ .

step 1.1L2algebra

3.1

By step 2.1, $g$ is a bounded entire function, so [L3] makes it constant.

step 2.1L3

4.1

The constant $g$ is nonzero, and $f = w + 1 / g$ is therefore constant, contradicting the given hypothesis; thus the image is dense.

step 3.1algebra∎

Depends on

Used by

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22 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)