I was wondering when a function was conformal at a pole? In class, when learning about Möbius transformations we put down a definition saying $f$ is conformal at a pole $z$ if $1/f$ is conformal at the zero $z$. However, I'm not sure whether this was given as a general statement, or whether it only applies to Möbius transformations.
Also, does it matter whether the function that has a pole is meromorphic, say, on the entire Riemann sphere, or does the answer change if it also has an essential singularity, such as, for example, $f(z) = \frac {1}{sinz}$ (pole at $z = 0$, but an essential singularity at $z = \infty$)?
I've looked online a bit, and found some sources saying that even Möbius transformations aren't conformal at their poles, i.e. at $z =-\frac{d}{c}$, which directly contradicts what I've learned. Unfortunately, the professor that is teaching the class is out of town for a week, so I can't ask him, and figured I'd turn to this community. It also seems it's somewhat of a general question that I haven't found answered anywhere else, so it might be good to hear more about.
For full disclosure, this is related to homework, but I feel the question is stated in general terms, not in terms of me trying to solve a specific problem (i.e. the problem on the homework isn't to answer this question directly), so I didn't attach a homework tag to it.