The bookUnit 2 · Analytic Functions31 / 83

Power series and analyticity

Why this is asked: Holomorphic ⇔ analytic ⇔ locally a convergent power series — an equivalence with no real-analysis analogue. Use the identity theorem to force f ≡ g from agreement on a set with a limit point *inside* the domain.

Analyticity, the identity theorem and zeros

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Why the radius of convergence is set by the nearest singularityinteractive

A function smooth on all of ℝ whose series stops at |x| = 1 — explained by a pole you cannot see on the real line.

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The trap here

“Zeros of a non-constant holomorphic function cannot accumulate” — false

sin(1/z) on $\mathbb{C}$ ∖{0}

Zeros $1/(n\pi)$ accumulate at 0, which lies outside the domain. Inside the domain zeros are always isolated.