Find and classify the singularity of the function

Question

For the complex function f(z), classify the singularity as either a pole, essential singularity or removable singularity.

$ f(z) = \frac{2z}{(z^2+1)(z-4)} $

I have expanded the bottom line to get $ \frac{2z}{z^3-4z^2+z-4} $. Then taking a factor of z from top and bottom $ \frac{2}{z^2-4z+1-\frac{4}{z}} $. Does this make it a pole of order 2?

Answer 1

You can read the singularities off the expression in the denominator: $$ (z-i)(z+i)(z-4) $$ they are simple: no power of $z$ in the numerator can help you at these poles by "cancelling" the blow up.