Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

$D(P,r)\backslash \{P\}\xrightarrow{f}U\xrightarrow{g}\mathbb{C}$. Both $f$ and $g$ are holomorphic. If $f$ has a removable (or pole, or essential) singularity at $P$, does $g\circ f$ also have one?

I think in any case $g\circ f$ will not have singularities any more. It is just a hunch, and I cannot explain it clearly.

share|improve this question
1  
You should try to look for potential functions $g$ that cancel out the singularities, and for functions that don't. (Hint: What happens if $g(x)=x$? What if $g$ is constant? And if the singularity of $f$ is removable, what happens if $g$ has a singularity at $f(P)$?) –  Lukas Geyer Oct 31 '12 at 18:53
    
Thank Lukas and Old John for your hints and details. If $f$ has essential singularity at $P$, by Casorati-Weierstrass theorem, $f(D(P,r)\backslash \{P\})=U$ is dense in $\mathbb{C}$. Does it mean $g$ has no singularity is this case? –  Sam Oct 31 '12 at 19:48
add comment

1 Answer 1

up vote 2 down vote accepted

A couple of things to consider:

If $g$ is the function $g:z \mapsto z$, then the composite will have exactly the same behaviour at $P$ as $f$ does.

At the other extreme, if $g$ is the constant function that maps everything to zero, then the composite function will have at worst a removable singularity, depending on whether you consider $\infty$ to be a point in the co-domain of $g$.

share|improve this answer
add comment

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.