Past exam of the mathematics course of the year 2 of the natural sciences course of the University of Cambridge / 2019 / Paper 2 / 4 / b

Table of contents
- Solution b

Solution

The Taylor series of the entire function

g

gives

\frac{g ( z )}{z - z _{0}} = \frac{g ( z _{0} )}{z - z _{0}} + g^{'} (z_{0}) + O (z - z_{0}) .

The coefficient of

(z - z_{0})^{- 1}

, and therefore the residue, is

Res_{z = z_{0}} \frac{g ( z )}{z - z _{0}} = g (z_{0}) .

If

g (z_{0}) = 0

, the apparent pole is a removable singularity and the residue is zero.

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