Past exam of the algebra, analysis and differential equations II course of the University of Oxford / 2008 printed variant / 5 / ii

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- Solution ii

Solution

Use

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

and

(1 - e^{- z})^{- 3} = z^{- 3} (1 - z /2 + O (z^{2}))^{- 3} = z^{- 3} (1 + 3 z /2 + O (z^{2})) .

Their product is

z^{- 2} + (3/2) z^{- 1} + O (1)

. Thus the singularity is a pole of order two, with

Res_{0} f = 3/2

.

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