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

Solution

D

is star-shaped with centre

z_{0}

: if

z \in D

, every point of

[z_{0}, z]

has its segment to

z_{0}

contained in the same segment and therefore avoids the origin. Applying part (a) and the stated triangular-integral fact to

1/ z

G (z) = \int_{[z_{0}, z]} \frac{d ζ}{ζ}, G^{'} (z) = \frac{1}{z}, G (z_{0}) = 0.

Choose either square root

c

z_{0}

h (z) = c e^{G (z) /2} .

Then

h

is holomorphic. Moreover,

\frac{d}{d z} (\frac{h ( z ) ^{2}}{z}) = 0,

and its value at

z_{0}

is one. Hence

h (z)^{2} = z (z \in D) .

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