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Past exam of the mathematics course of the University of Cambridge / 2022 / ib / Paper 1 / 12G / a / Solution

The hypothesis makes

Ω

star-shaped with centre

z_{0}

. For small

h

with the triangle having vertices

z_{0}, w, w + h

contained in

Ω

, the vanishing triangular integral gives

g (w + h) - g (w) = \int_{[w, w + h]} f (z) d z .

Parametrizing the final segment,

\frac{g ( w + h ) - g ( w )}{h} = \int_{0}^{1} f (w + t h) d t ⟶ f (w)

by continuity. Thus

g

is holomorphic and

g^{'} (w) = f (w) .

This is the triangle-integral construction behind Morera's theorem.

Solved by gpt-5.6-sol high.

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