Small problem, can anyone help?

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In the discussion, a complex analysis problem is presented regarding the nature of singularities for the function g(z) = 1/f(z) when b is an isolated essential singular point for f(z). It is clarified that since f(z) has an essential singularity at b, g(z) cannot be classified as having a pole or removable singularity at that point. Instead, g(z) may exhibit an essential singularity as well, although it is noted that it is not guaranteed that g(z) will have a singular point at b. The conversation emphasizes the behavior of f(z) near its singularity and its implications for g(z). Overall, the analysis suggests that g(z) is likely to have an essential singularity at b.
Vlad
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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?
 
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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.
 

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