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If I look at the function ## (z^2+z-2)/(z-1)^2## it appears to have a double pole at z=1 but if I factorise the numerator I get ##z^2+z-2 = (z+2)(z-1)## and it is a simple pole.

Is it wrong to say it is a double pole ?

If I overestimate the order of the pole in this case as 2 and calculate the residue using limits and differentials I still get the correct answer. Is this always true ?

In this case the numerator was easy to factorise. If it was a complicated function involving higher powers that couldn't be factorised is it possible to say for certain what the order of the pole is just by looking at the denominator ?

Thanks