The bookUnit 2 · Cauchy Theory33 / 83

Liouville, Morera, maximum modulus principle

Why this is asked: Recognise Liouville in disguise: bounded, bounded real part, bounded on a growth scale, or missing two values. Maximum modulus turns interior bounds into boundary bounds.

Liouville, Morera and the maximum principle

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Liouville, and the growth bounds that do not force constancyinteractive

Four hypotheses — which collapse the function to a constant and which only make it a polynomial.

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The trap here

“A bounded holomorphic function on an unbounded domain is constant” — false

f(z) = eᶻ on {Re z < 0}

|eᶻ| < 1 there, but f is not constant. Liouville needs the whole plane.