Recently we have been covering basic complex analysis concepts in my class and one of the things I'm struggling with is how to determine if a function is analytic at either infinity or zero. For instance, consider the following functions:
$f_1(z)=\frac{sin(z)}{z} \rightarrow f_1'(z)=\frac{cos(z)}{z}-\frac{sin(z)}{z^2}$
$f_2(z)=\frac{1}{z^2+1} \rightarrow f_2'(z)=\frac{-2z}{(z^2+1)^2}$
For $f_1$ I would have thought that it is analytic everywhere besides $0$ and $\infty$ but they are saying that it is analytic everywhere except $\infty$. For $f_2$ I was thinking again that the analytic domain would just be excluding $0$, $\infty$ , and $\pm i$ but according to the text they are claiming it is just $\pm i$. What exactly do you look for when looking for points where a function is not analytic?