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

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The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

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$\ker(\exp)=2\pi i\mathbb Z$ , and $\exp z=\exp w$ exactly when $z-w\in2\pi i\mathbb Z$

Statement

$\ker(\exp)=2\pi i\mathbb Z,$ and $\exp z=\exp w$ exactly when $z-w\in2\pi i\mathbb Z$ . The conventions and prerequisite facts used below are recorded in $\exp(x+iy)=e^x(\cos y+i\sin y)$ , $|\exp(x+iy)|=e^x$ , and $e^{i\pi}+1=0$ , The exponential function is strictly increasing, The zero sets of sine and cosine and the least positive common period 2 pi, $t\mapsto(\cos t,\sin t)$ is a bijection from $[0,2\pi)$ onto the real unit circle.

Facts & Assumptions

Given: $z=x+iy$ and $w=u+iv$ .

Proof

technique · direct

1.1

Cartesian form shows $\exp z=1$ forces $e^x=1$ and $(\cos y,\sin y)=(1,0)$ .

given

1.2

Strict monotonicity gives $x=0$ , while the trigonometric period theorem gives $y\in2\pi\mathbb Z$ .

given

2.1

The addition law turns equality of two exponential values into membership of $z-w$ in the kernel, and the converse is immediate.

given∎

Depends on

Used by

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 86 results over 22 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)

ker⁡(exp⁡)=2πiZ\ker(\exp)=2\pi i\mathbb Zker(exp)=2πiZ, and exp⁡z=exp⁡w\exp z=\exp wexpz=expw exactly when z−w∈2πiZz-w\in2\pi i\mathbb Zz−w∈2πiZ