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Weierstrass M-test for complex-valued function series
Statement
Let be a set and . Suppose , for all , and the real series converges. Then converges absolutely for every and its partial sums converge uniformly on in the sense of Uniform convergence and the uniformly Cauchy condition for complex-valued functions, with the componentwise dictionary.
Facts & Assumptions
Given: Functions and a convergent nonnegative majorant series as in the Statement.
A complex-valued function sequence converges uniformly if and only if it is uniformly Cauchy (A sequence of complex-valued functions converges uniformly if and only if it is uniformly Cauchy).
The real Weierstrass M-test states that the same majorant hypotheses give absolute pointwise and uniform convergence for real-valued functions (The Weierstrass M-test gives absolute pointwise convergence and uniform convergence of a function series).
Proof
For partial sums and , [L1] gives for every .
Since the real series is Cauchy, its tails make the bound in step 1.1 uniformly small; thus is uniformly Cauchy and converges uniformly by [L2].
For each , the nonnegative series is bounded termwise by , exactly the comparison used in [L3], and therefore converges. Zero majorants and the empty set require no separate choice.
Depends on
- Uniform convergence and the uniformly Cauchy condition for complex-valued functions, with the componentwise dictionary
- A sequence of complex-valued functions converges uniformly if and only if it is uniformly Cauchy
- The Weierstrass M-test gives absolute pointwise convergence and uniform convergence of a function series
- Conjugation is an involutive real-field automorphism, $z\overline z=|z|^2$, and modulus is definite, multiplicative, and subadditive
Used by
- The series sum z to the n over n squared is continuous on the closed disc but singular at 1 Counterexample
- Uniform convergence on the closed unit disc does not give a holomorphic extension to a larger disc Counterexample
- Multi-indexed power series in ℂᵐ and their absolute convergence Definition
- A power series with reciprocal-square coefficients converges uniformly on the closed unit disc Example
- The Cauchy kernel expands as an absolutely and uniformly convergent multi-indexed geometric series Lemma
- The Dirichlet series for zeta converges absolutely and locally uniformly on the half-plane Res>1 Lemma
- Trace-norm continuity, growth and multiplicativity of the local determinant Lemma
- Products of convergent complex power series are represented by their Cauchy-product coefficients on the common disc Proposition
- A complex power series converges absolutely and uniformly on every closed subdisc strictly inside its disc of convergence Theorem
- A continuous separately holomorphic function is the sum of an absolutely convergent power series with Cauchy-integral coefficients on every smaller polydisc Theorem
- Absolute convergence at one point forces absolute and locally uniform convergence on closed half-planes to the right Theorem
- An absolutely convergent multi-indexed power series is holomorphic and differentiates termwise Theorem
- The Dirichlet eta series is holomorphic on Res>0 and equals the prefactor times zeta there Theorem
Dependency tree · two levels
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Sources
- L. Ahlfors, Complex Analysis, 3rd ed., Ch. 2 (standard reference, not scraped)