I am working on a textbook exercise. A similar question: An analytic function in a compact region has finitely many zeros, but it's not quite clear to me and I also have possibly another approach? I want to prove basically the same question, that if $f$ is analytic inside and on a simple closed contour $C$ (except possibly for poles inside $C$), and if all zeros of $f$ are inside $C$ and of finite order, then the zeros must be finitely many.
Hopefully my attempt below can be verified or corrected.
Suppose otherwise. Then by Bolzano-Weierstrass, the set $S$ of all zeros of $f$ (which is infinite) contains an accumulation point inside $C$. Let's say it is $z_0$. This $z_0$ is also a zero of $f$ since it is the limit a subsequence of zeros in $S$ and $f$ is analytic (hence continuous too). By assumption, it is a zero of finite order, say $m$.
I claim that in any neighborhood $N$ of $z_0$, $f$ cannot be identically zero. To see this, write $f(z)=(z-z_0)^mg(z)$ where $g$ is nonzero and analytic at $z_0$. Hence by these properties of $g$, there is a neighborhood around $z_0$ (intersected with $N$) where $g$ is nonzero. However, this neighborhood contains another (different) zero, say $z'$, of $f$ by definition of accumulation point. Hence, $0=f(z')=(z'-z_0)^mg(z')$, implying that $g$ can be zero in this neighborhood, a contradiction.
Now by a theorem in the textbook, since $f$ is analytic and zero at $z_0$, but not identically zero in any neighborhood of $z_0$, there must be a deleted neighborhood of $z_0$ where $f$ is identically nonzero. But again, in this deleted neighborhood contains a zero of $f$, say $z''$, by definition of accumulation point, contradicting $f$ being identically nonzero there. QED.
So my questions would be:
Is the above valid? If not, which part should be improved?
Are there any other approaches?
Usually Q2 is more interesting, but I highly appreciate if Q1 is answered too. Thanks a lot!
EDIT: Now that I think about it after some comment inputs:
My first paragraph should be fine.
- As for my second paragraph until conclusion, I should do it like this:
As $z_0$ is of order $m$, we can write $f(z) = (z-z_0)^m g(z)$ where $g$ is analytic and nonzero at $z_0$. By continuity of $g$ and being nonzero at $z_0$, there is a neighborhood at $z_0$ where $g$ is identically nonzero. Deleting $z_0$ there, $f$ is then nonzero in that deleted neighborhood. However, this contradicts the fact that $z_0$ is an accumulation point of zeros. Done?
- Another method, I can also say: Either $f$ is not identically zero in any neighborhood $N$ of $z_0$ , or $f$ is identically zero in some neighborhood $N$ of $z_0$ . For the former, my original third paragraph follows to conclude. For the latter, by identity theorem $f$ must be identically zero inside $C$. By analyticity, their derivatives of all order are zero, showing infinite order. Done?