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Mathematics course of the University of Cambridge

Past exam of the mathematics course of the University of Cambridge

2019

Paper 2

For $|z|<1$, sum the geometric series and its derivative:
$$
\sum_{n\geq0}(-1)^nz^n=\frac1{1+z},
\qquad
\sum_{n\geq0}n(-1)^nz^n=-\frac z{(1+z)^2}.
$$
Therefore
$$
h(z)=\frac1{1+z}-\frac{2z}{(1+z)^2}
=\frac{1-z}{(1+z)^2}.
$$
This rational continuation is holomorphic across every point of the unit circle except $z=-1$, where it has a double pole. The set of singular points is consequently
$$
\boxed{\{-1\}}.
$$

Solution

Past exam of the mathematics course of the University of Cambridge / 2019 / ii / Paper 2 / 23F / b / Solution

Past exam of the mathematics course of the University of Cambridge / 2019 / ii / Paper 2 / 23F / b / Solution

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Past exam of the mathematics course of the University of Cambridge / 2019 / ii / Paper 2 / 23F / b / Solution

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