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.
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Every nonzero complex number has a unique polar form with and
Statement
Every has a unique representation with and . The conventions and prerequisite facts used below are recorded in Real and imaginary parts, complex conjugation, and modulus, is a bijection from onto the real unit circle, The zero sets of sine and cosine and the least positive common period 2 pi.
Facts & Assumptions
Given: .
Proof
The point lies on the unit circle.
The unit-circle parametrization supplies an angle, and its endpoint convention converts it uniquely to .
Multiplying by proves existence; the period theorem proves uniqueness.
Depends on
Used by
- The principal logarithm is the normalised holomorphic branch on the slit plane Corollary
- A curl-free C¹ field on the complement of a line that is not conservative Counterexample
- A holomorphic function on an annulus can have a nonzero closed-contour integral Counterexample
- Complex logarithms, the principal logarithm, and principal and multivalued complex powers Definition
- Canonical Banach complexification of a real Banach space Lemma
- The complex exponential maps ℂ onto ℂ∖{0} Theorem
- The n-th roots of a complex number and the n distinct roots of unity for every n≥1 Theorem
- Trigonometric integrals become contour integrals by the unit-circle substitution Theorem
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)