Past exam of the mathematics course of the year 2 of the natural sciences course of the University of Cambridge / 2019 / Paper 1 / 7 / d / i

Table of contents
- Solution i

Solution

The denominator factors as

z^{2} + i z + 6 = (z - 2 i) (z + 3 i)

, with no cancellation against the numerator. Thus there are

simple poles at z = 2 i, - 3 i; a simple zero at z = 4.

To inspect infinity, put

w = 1/ z

:

f (1/ w) = \frac{w - 4 w ^{2}}{1 + i w + 6 w ^{2}} = w + O (w^{2}) .

Hence infinity is a removable singularity at infinity, with limiting value zero. After this natural extension to the Riemann sphere it is also a simple zero. There are no other zeros or singularities.

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