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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The complex exponential series converges absolutely for every complex argument
Statement
For every , the series converges absolutely. The conventions and prerequisite facts used below are recorded in The complex exponential by its power series, The exponential series converges absolutely for every real argument, Every absolutely convergent complex series converges, and rearrangements preserve its sum.
Facts & Assumptions
Given: .
Proof
Its modulus series is , the real exponential series at the nonnegative real .
The real infinite-radius lemma and complex absolute-convergence theorem give the result.
Depends on
Used by
- Characteristic functions of bernoulli binomial and poisson laws Example
- Independent sums via characteristic functions Example
- The complex exponential satisfies the Cauchy–Riemann equations in Cartesian and polar form Example
- The power series of exp(z₀+z₁) on every bidisc Example
- The unit-circle integral of exp(z)/z is 2 pi i by uniform termwise integration Example
- Products of near-one characteristic factors Lemma
- Euler's formula: exp(iθ)=cosθ+i sinθ for every real θ Theorem
- exp(z+w)=exp z exp w, and the complex exponential extends the real exponential Theorem
- Gleason Kahane Zelazko Theorem
- The exponential definitions of complex sine, cosine, hyperbolic sine, and hyperbolic cosine equal their entire power series Theorem
Cited to discharge well-definedness by The complex exponential by its power series.
Dependency tree · two levels
16 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)