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.
Complex sine, cosine, hyperbolic sine, and hyperbolic cosine from the complex exponential
Definition
For , define The conventions and prerequisite facts used below are recorded in The complex exponential by its power series, is a field, every element is uniquely , and every nonzero element has inverse .
Depends on
Used by
- The addition formulas for complex trigonometric and hyperbolic functions Corollary
- 1/sin(1/z) has a nonisolated singularity at 0 Counterexample
- sin(1/z) has an essential singularity at 0 Counterexample
- 1/z is not uniformly approximable by polynomials on the unit circle Example
- Complex sine and cosine are unbounded on the complex plane Theorem
- The exponential definitions of complex sine, cosine, hyperbolic sine, and hyperbolic cosine equal their entire power series Theorem
- The exponential formulas, real restrictions, and trigonometric-hyperbolic dictionary over ℂ Theorem
- The Mittag-Leffler expansion of pi cotangent Theorem
- The sine map biholomorphically sends an upper half-strip onto the upper half-plane Theorem
- The Weierstrass product for sine Theorem
- The zeros of complex sine are the integer multiples of pi, and the zeros of complex cosine are the odd half-integer multiples of pi Theorem
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)