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

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$1 = e^{2 π i}$ does not imply $0 = 2 π i$ : logarithms invert the exponential only modulo its kernel

Example

Although $1 = \exp (2 π i)$ , one may not conclude $0 = 2 π i$ : logarithms classify fibres only modulo $2 π i Z$ . The conventions and prerequisite facts used below are recorded in $\exp (x + i y) = e^{x} (\cos y + i \sin y)$ , $∣ \exp (x + i y) ∣ = e^{x}$ , and $e^{i π} + 1 = 0$ , $\ker (\exp) = 2 π i Z$ , and $\exp z = \exp w$ exactly when $z - w \in 2 π i Z$ , All logarithms of $z \neq 0$ are $Log z + 2 π i k$ , $k \in Z$ .

Facts & Assumptions

Given: The two complex numbers $0$ and $2 π i$ .

Verification

1.1

Both lie in the fibre of $1$ , and their difference is the nonzero kernel element $2 π i$ .

given

2.1

Thus equality of exponential values identifies logarithms only modulo the kernel, not as equal complex numbers.

algebra∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

15 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)

1=e2πi1=e2πi does not imply 0=2πi0=2πi: logarithms invert the exponential only modulo its kernel