ExampleConstruction: AI-adaptedVerification: AI-adaptedprecheck passaudited 2026-08-02

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All values of $i^{i}$ are the positive real numbers $e^{- π / 2 - 2 π k}$ , $k \in Z$

Example

All values of $i^{i}$ are $e^{- π / 2 - 2 π k}$ for $k \in Z$ , and each is a positive real number. The conventions and prerequisite facts used below are recorded in Complex logarithms, the principal logarithm, and principal and multivalued complex powers, All logarithms of $z \neq 0$ are $Log z + 2 π i k$ , $k \in Z$ .

Facts & Assumptions

Given: $Log i = i π / 2$ .

Verification

1.1

The logarithms of $i$ are $i (π / 2 + 2 π k)$ .

given

2.1

Multiplication by $i$ gives the real exponents $- π / 2 - 2 π k$ , whose exponentials are positive.

algebra∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

8 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

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

All values of iiii are the positive real numbers e−π/2−2πke−π/2−2πk, k∈Zk∈Z