Suppose I have the function:
$ f(z) = \frac{log(z+1)}{z^3} $
I need to find the Laurent series of this function to determine its order.
In this case we take $log(z)$ to be $ln|z|+i\arg (z)$
So we want to find the power series for $\frac{ln|z+1|}{z^3} + \frac{i \arg (z+1)}{z^3}$
$ = \sum_{n=1}^{\infty}[\frac{(-1)^{n+1}z^{n-3}}{n} ]+ \frac{i \arg(z+1)}{z^3}$
I'm not sure how to incorporate the arg into the series though, any ideas?
EDIT: Log defined in my notes
One may recall that, as $z \to 0$, by using the Taylor series expansion, $$ \log(1+z)=z-\frac{z}2+\frac{z^3}3+O(z^4) $$ giving the Laurent series expansion $$ \frac{\log(1+z)}{z^3}=\frac1{z^2}-\frac1{2z}+\frac13+O(z) $$as $z \to 0$.