TheoremStatement: AI-adaptedProof: AI-adaptedSession-authored (Fable 5 assisted)precheck passaudited 2026-08-02

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The complex exponential maps $\mathbb C$ onto $\mathbb C\setminus\{0\}$

Statement

The complex exponential maps $\mathbb C$ onto $\mathbb C\setminus\{0\}$ . The conventions and prerequisite facts used below are recorded in Every nonzero complex number has a unique polar form $r(\cos\theta+i\sin\theta)$ with $r>0$ and $-\pi<\theta\le\pi$ , $\exp(x+iy)=e^x(\cos y+i\sin y)$ , $|\exp(x+iy)|=e^x$ , and $e^{i\pi}+1=0$ , The natural logarithm as the inverse of the exponential function, Order, continuity, range, and the product, quotient, and reciprocal laws for the natural logarithm.

Facts & Assumptions

Given: $z\in\mathbb C\setminus\{0\}$ .

Proof

technique · constructive

1.1

Write $z=r(\cos\theta+i\sin\theta)$ with $r>0$ .

given

1.2

Put $w=\log r+i\theta$ .

construct

2.1

Cartesian exponential form and $e^{\log r}=r$ give $\exp w=z$ .

discharge-construct∎

Depends on

Used by

All logarithms of z≠0 are Logz+2π i k, k∈ℤ Theorem

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 70 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

J. Lebl, Basic Analysis I: Complex Numbers and the Complex Exponential (standard reference, not scraped)

The complex exponential maps C\mathbb CC onto C∖{0}\mathbb C\setminus\{0\}C∖{0}