Past exam of the metric spaces and complex analysis course of the University of Oxford / 2014 / 5 / a

Write the Laurent series about

a

\sum_{k \in Z} c_{k} (z - a)^{k}

removable singularity if every negative-index coefficient is zero;
a pole of order $m \geq 1$ if $c_{- m} \neq = 0$ and $c_{k} = 0$ for $k < - m$ ;
an essential singularity if infinitely many negative-index coefficients are nonzero.

These cases exhaust the possible principal parts. The residue is

res (f; a) = c_{- 1} = \frac{1}{2 πi} \int_{∣ z - a ∣ = ρ} f (z) d z, 0 < ρ < r .

For the first example,

sin z = z + O (z^{3})

gives

\frac{1}{s i n ^{2} z} = \frac{1}{z ^{2}} + \frac{1}{3} + O (z^{2}) .

Thus zero is a double pole, with residue zero. For the other example,

1 - cos z = \frac{z ^{2}}{2} + O (z^{4}), 1 - e^{z} = - z + O (z^{2}),

so the ratio is

- z /2 + O (z^{2})

. Its singularity is removable, with extended value zero and residue zero. The numerator's zero has higher order than the denominator's, so there is no pole.

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