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The complex exponential series converges absolutely for every complex argument
Statement
For every , the series converges absolutely.
Facts & Assumptions
Given: .
A complex series converges absolutely when its real modulus series converges (Complex series, absolute convergence, complex power series, and radius of convergence).
Complex modulus is multiplicative, and natural factorials are positive and nonzero (Conjugation is an involutive real-field automorphism, , and modulus is definite, multiplicative, and subadditive, Integer powers in the complex field, The factorial and the falling factorial , defined by recursion in , Canonical naturals are positive and strictly increasing).
For every real , the real series converges absolutely (The exponential series converges absolutely for every real argument).
Absolute convergence of a complex series implies convergence (Every absolutely convergent complex series converges, and rearrangements preserve its sum).
Proof
Each factorial denominator is nonzero, and multiplicativity of modulus gives . Thus its modulus series is the real exponential series at the nonnegative real .
By [F3] this modulus series converges; [F1] makes the complex series absolutely convergent, and [F4] gives convergence as well.
Depends on
- Complex series, absolute convergence, complex power series, and radius of convergence
- Integer powers in the complex field
- The factorial $n!$ and the falling factorial $n^{\underline{k}}$, defined by recursion in $\mathbb{N}$
- Canonical naturals are positive and strictly increasing
- Conjugation is an involutive real-field automorphism, $z\overline z=|z|^2$, and modulus is definite, multiplicative, and subadditive
- The exponential series converges absolutely for every real argument
- Every absolutely convergent complex series converges, and rearrangements preserve its sum
Used by
- Hamiltonian-simulation polynomial target Definition
- The Bernoulli numbers are defined by the generating series t/(eᵗ-1) Definition
- The complex exponential by its power series Definition
- 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
- Completeness of normalized probabilists’ Hermite polynomials in Gaussian L² Lemma
- Error-function series geometric tail bound Lemma
- Integral representations and a range bound for integer Bessel functions Lemma
- Jacobi–Anger Chebyshev truncation Lemma
- Products of near-one characteristic factors Lemma
- Correctness and arithmetic resource bound for buffered phase approximation Theorem
- 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
- Soundness and eventual absolute precision for spherical Bessel enclosures Theorem
- Taylor simulation tail bound Theorem
- The exponential definitions of complex sine, cosine, hyperbolic sine, and hyperbolic cosine equal their entire power series Theorem
Dependency tree · two levels
45 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)