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Working on a problem in Gamelin's book.

"Show that if $f(z)$ is a continuous function on a domain $D$ such that $f(z)^N$ is analytic on $D$ for some integer $N$, then $f(z)$ is analytic on $D$."

He starts a hint with the statement "Show that the zeros of $f(z)$ are isolated." This makes me think. I know that if $f$ is analytic and nonconstant, then the zeros are isolated. Is there some sort of converse to this theorem? I.e., for example, if continuous and zeros are isolated then analytic?

His next hint is: "At a zero of $f(z)$, write $f(z)^N=(z-z_0)^mh(z)$ where $h(z_0)\ne 0$ and show that $N$ divides $m$." However, the book has covered nothing on winding numbers thus far. Also, how will the fact that $N$ divides $m$ help to show that $f(z)$ is analytic?

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" if continuous and zeros are isolated then analytic?" This can't be true. There are many examples which are not analytic and continuous and zeros are isolated. Consider $f(z)=|(z-1)(z-2)(z-3)|$ –  i707107 Apr 7 '13 at 21:28
Good counterexample. –  David Apr 8 '13 at 0:33

2 Answers 2

Strictly speaking, one should dispose of $f\equiv 0$ first. After that, we know that the zeros of $f$ are isolated, because they are precisely the zeros of $f^N$. Let $Z$ be the set of zeros of $f$. The answer by Harald Hanche-Olsen explains how to show that $f$ is holomorphic in $D\setminus Z$. To finish off the problem, it is not necessary to follow the hint about $N$ dividing $m$. Instead one can use the following

Fact. A function that is continuous in $D$ and holomorphic in $D\setminus Z$ (where $Z$ is a discrete set) is holomorphic in $D$.

Sketch of proof: for any $a\in Z$ the Laurent series of $f$ around $a$ does not contain any negative powers, since $f$ is bounded in a neighborhood of $a$. (Use the Cauchy estimate for coefficients, for example).

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Write $g(z)=f(z)^N$. Away from the zeros, $f(z)$ is locally of the form $g(z)^{1/N}$, where the appropriate branch of the $N$th root is (locally) consistent. This should give you analyticity away from the zeros.

For the last part, a knowledge of winding numbers is certainly helpful. As for the conclusion from knowing that $N\mid m$, think back to those $N$th roots. You will find that, as you go around the zero, you can choose the $N$th root in a continuous manner all the way, getting back to the same branch after a full circle.

At least this is how I would think about it. I can't say if it is appropriate given what you do and don't know at that stage in the book.

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