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Counterexample bank

Part C is won by knowing which tempting claims are false. 148 counterexamples; 85 free. The rest come with the Notes pack.

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#1 · Complex Analysis, Algebra & Topology › Singularities and Residues

“|f| bounded near an isolated singularity ⇒ pole” — false

Counterexample: f(z) = sin(z)/z at 0

Bounded ⇒ removable (Riemann). Poles have |f| $\to \infty$ .

complexsingularities

#2 · Complex Analysis, Algebra & Topology › Singularities and Residues

“If |f| is bounded near an isolated singularity, the singularity is a pole” — false

Counterexample: f(z) = sin z / z at 0

Bounded ⇒ removable (Riemann). Poles have |f| $\to \infty$ .

complexsingularities