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

Example

All values of $i^i$ are $e^{-\pi/2-2\pi k}$ for $k\in\mathbb Z$ , 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 $z\ne0$ are $\operatorname{Log}z+2\pi i k$ , $k\in\mathbb Z$ .

Facts & Assumptions

Given: $\operatorname{Log}i=i\pi/2$ .

Verification

1.1

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

given

2.1

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

algebra∎

Depends on

Used by

Nothing in the library uses this result yet.

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Sources

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

All values of iii^iii are the positive real numbers e−π/2−2πke^{-\pi/2-2\pi k}e−π/2−2πk, k∈Zk\in\mathbb Zk∈Z