ExampleConstruction: AI-adaptedVerification: AI-adaptedSession-authored (Fable 5 assisted)precheck passaudited 2026-08-02

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The logarithms of $-1$ are $(2k+1)\pi i$ , $k\in\mathbb Z$

Example

The logarithms of $-1$ are $(2k+1)\pi i$ for $k\in\mathbb Z$ . The conventions and prerequisite facts used below are recorded in All logarithms of $z\ne0$ are $\operatorname{Log}z+2\pi i k$ , $k\in\mathbb Z$ , $\exp(x+iy)=e^x(\cos y+i\sin y)$ , $|\exp(x+iy)|=e^x$ , and $e^{i\pi}+1=0$ .

Facts & Assumptions

Given: $\operatorname{Log}(-1)=i\pi$ .

Verification

1.1

Add the kernel $2\pi i\mathbb Z$ to the principal logarithm.

given

2.1

Simplifying $i\pi+2\pi ik$ gives $(2k+1)\pi i$ for every integer $k$ .

algebra∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 42 results over 11 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 logarithms of −1-1−1 are (2k+1)πi(2k+1)\pi i(2k+1)πi, k∈Zk\in\mathbb Zk∈Z