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QUESTION


(From 1996 exam.) Classify the singularities of the function
$$cos(z\pi/2)sin(z^{-1})\over(z^2-1)(z-2).$$



ANSWER


The points $z$ where singularities may occur are $z=0, 1,-1, 2$.
When $z=0$ there must be an essential singularity as $\sin
(z^{-1})$ has a Laurent expansion about $z=0$ with an infinite
number of negative powers of $z$. At $z=2$ there is a simple pole.
As$\cos (\pi/2)=\cos (-\pi/2)=0$ w need to take care of the points
$z=1$ and $z=-1$. If, for example, we use L'H\^opital's rule, we
find that the limit as $z\rightarrow 1$ of ${\cos (z\pi/2)\over
z^2-1}$ is finite and also for the limit as $z\rightarrow -1$.
Thus there are removable singularities when $z=1$ and $z=-1$.




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