For each of the following functions classify the isolated singularity at 0 and specify the principal part of the Laurent development there:

a) $\dfrac{sin(z)}{z^n},\;n\in\mathbb{N}$

b) $\dfrac{z}{(z+1)sin(z^n)},\;n\in\mathbb{N}$

c) $cos(z^{-1})sin(z^{-1})$

d) $(1-z^{-n})^{-k},\;n,k\in\mathbb{N}\setminus\{0\}$

I think that in a) $0$ is a removable singularity, in b,c and d $0$ is an essential singularity, but what does it say specify the principle part of the Laurent development? How do I do it?

  • $\begingroup$ It might help you to write out the first two or three terms of the Maclaurin series for sine. This will make it easier to see what happens in the first case when $n > 1$. $\endgroup$ – Eric Towers Apr 24 at 16:56
  • $\begingroup$ In a) $0$ is a pole for $n\gt 1$. $\endgroup$ – Thomas Shelby Apr 24 at 16:57

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