Re: -- Quickie: finite number of complex zeros in neighbourhood

On Feb 8, 6:33�am, Angus Rodgers <twir...@xxxxxxxxxxx> wrote:
I was reading some textbooks in 2004, but ran out of steam. �I've
picked one of them up again, but I'm stuck at the beginning of a
proof which (according to my notes) caused me no trouble in 2004.

There's nothing essentially wrong with the proof, and I can't see
anything wrong with my reasoning either; the problem is just that
I can't imagine what the author has in mind (nor what I imagined
he had in mind when I read the proof in 2004).

He has defined an analytic function of a complex variable as one
that possesses a local power series expansion at every point of
its domain. �He has proved that if such a function is not zero
everywhere, its zeros are isolated. �Having deduced the Maximum
Modulus Theorem, Liouville's Theorem, Schwarz's Lemma, and Cauchy's
Inequalities (all easily, and without using complex differentiation
or integration), he now wants to prove a version of the Argument
Principle. �For this, he needs to show that an analytic function
not zero everywhere has only finitely many zeros in any compact
subset, K, of its domain, D. �Without yet using compactness (this
comes in the next step), he writes:

�As f is not constant, each zero of f is isolated and so each z
�in D is the centre of some open disc which contains only a finite
�number of zeros of f.

Now, it is obvious that if f(z) = 0, then the known fact that z is
isolated means that it is the centre of a disc containing no other
zeros. �And on the other hand, if f(z) != 0, then the known fact
that f is continuous at z means that z is the centre of a disc
containing no zeros at all. �So his statement is obviously true
(and immediately leads to the required result about K). But what
argument did the author have in mind that leads only to the number
of zeros in the disc around z being finite, rather than it always
being 0 or 1? (Of course, the latter fact is an obvious consequence
of the former.)

This is driving me nuts! �Normally when I fail to understand some-
thing in a book, at first I feel like an idiot, but then I quickly
find either that I have made a stupid mistake, which I can correct,
or else that there is an error in the text, which I can also correct
(sometimes only after some hard work), and than I no longer feel like
an idiot, and I can continue confidently. �But this time I'm stuck
with feeling like an idiot! �I hope old age hasn't caught up with me
at last. �Can someone give me a clue, and put me out of my misery?

--
Angus Rodgers
(twirlip@ eats spam; reply to angusrod@)
Contains mild peril

I usually prove this without the topological results: if every disk
about z contains an infinite number of zeros of f(z), then, since f
has derivatives of all orders at z (from the existence of the power
series). The given condition implies that all of the derivatives at z
are 0, so f is constant in the disk.
.

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