Alphabeta MathHow statement and proof provenance work
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.
These labels describe origin, not correctness: citations and verification chips remain separate evidence.
All logarithms of are ,
Statement
For , the solutions of are exactly for . The conventions and prerequisite facts used below are recorded in Complex logarithms, the principal logarithm, and principal and multivalued complex powers, , and exactly when , The complex exponential maps onto .
Facts & Assumptions
Given: and .
Proof
The principal logarithm is one solution by its polar definition.
Equality is equivalent to by the fibre theorem.
Depends on
Used by
- The principal logarithm is the normalised holomorphic branch on the slit plane Corollary
- ((-1)²)^1/2=1≠-1=(-1)^2(1/2) for principal complex powers Counterexample
- 1=e^2π i does not imply 0=2π i: logarithms invert the exponential only modulo its kernel Example
- All values of iⁱ are the positive real numbers e^-π/2-2π k, k∈ℤ Example
- The logarithms of -1 are (2k+1)π i, k∈ℤ Example
- A nonvanishing holomorphic function on a disc has a holomorphic logarithm Lemma
Dependency tree · two levels
12 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)