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.

 

[tacman]

Sure, I can try to help you with this problem. First, let's define some terms. An isolated essentially singular point is a point where the function is not defined, but the function approaches a finite value as we approach that point. Now, for g(z) = 1/f(z), if b is an isolated essentially singular point for f(z), then g(z) will also have an isolated essentially singular point at b. This is because g(z) is essentially the reciprocal of f(z), so if f(z) approaches a finite value at b, then 1/f(z) will also approach a finite value at b. Therefore, g(z) will have the same type of singularity as f(z) at b. It will not be a pole or a removable singularity, but an isolated essentially singular point. I hope this helps! Let me know if you need any further clarification.