Small problem, can anyone help?

[Vlad]

Just got this complex analysis problem that's bugging me. If b in U is an isolated essentially singular point for f(z) in U, what type of singularity can
g(z) = 1/f(z) have? Is it just an essentially singular pt for g(z) as well, it's not a pole or removable singularity is it? Can anyone help me with this?

[Gokul43201]

I believe you're right. Since U is an essential singularity, f(z) need not approach infinity, as z approaches U. So, if f(z) = w at U, then g(z) will not, in general, go to infinity or to zero. Of course, it's not clear that g(z) must even have a singular point at U, but it looks like it will.