Re: -- Quickie: finite number of complex zeros in neighbourhood
- From: Angus Rodgers <twirlip@xxxxxxxxxxx>
- Date: Sat, 09 Feb 2008 00:07:31 +0000
On Fri, 08 Feb 2008 13:25:03 -0800, The World Wide Wade
<aderamey.addw@xxxxxxxxxxx> wrote:
In article <95eoq311gniobgfe6rvlslalokg8b5ae7t@xxxxxxx>,
Angus Rodgers <twirlip@xxxxxxxxxxx> wrote:
[...] 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.
[...]
He might be confusing the order of a zero at a point and the number of
distinct points where f is 0. At any rate he's a sloppy author.
He's actually very careful, with a great sense of elegance, and
he obviously cares a lot about finding a natural and economical
development of a topic, from first principles (all qualities I
value).
If I do have a general problem with his style (apart from the
odd glitch like this one - I think there's been only one other
such case in the book so far, although there's about the usual
number of typos, easily correctable false statements, and so on)
it is that he tends sometimes to slip in statements of results
in a casual manner (but not sloppy - I mean without any fanfare)
which belies their importance, and may cause the naive reader
(i.e. me!) to fail to commit them to memory. (To compensate for
this, I've been preparing a summary of results established so far.
Ideally it shouldn't be necessary to do this, and one should be
able to see what has been established by casting an eye quickly
over the main headings of theorems, definitions, and so on.)
Perhaps the most confusing example of this, for me, is his treat-
ment of the fact that if f has valency [is that the standard
term?] s = v_f(z) at z, then for sufficiently small r > 0, the
winding number about the point f(z) of the curve f o g, where
g(t) = z + e^{it} (0 <= t <= 2pi), is s. The reader more or less
has to infer this for himself from some brief remarks on page 119,
then it is stated explicitly on page 125 - but not as a named or
numbered proposition - then it is used in an essential way but
without any formal reference (just a brief restatement without
proof, as if it were trivial) on page 129.
I think the explanation for this is that the author thinks it is
a simple, basic, and by now for him utterly obvious fact, and he
assumes the reader is either very sharp (I'm quite sharp but not
quite sharp enough) or already experienced or sophisticated (I'm
neither!), and can pick these things up without having them
hammered in by the use of names, numbers, bold text, or italics
- especially as he has already given a slower and more explicit
treatment for the case of polynomials, to prepare the ground.
(The whole thing is very well-planned; it's just the detailed
execution that I sometimes find confusing.)
Having now said so much about it, I should say what book it is!
It's one I described as "thought-provoking" in another recent
thread: A. F. Beardon, "Complex Analysis", Wiley (1979). I find
it quite tough going, but fascinating, at least when I'm capable
of concentrating. I'll get to Cauchy's Theorem if it kills me;
and after that, it's a long slog to the Jordan Curve Theorem.
--
Angus Rodgers
(twirlip@ eats spam; reply to angusrod@)
Contains mild peril
.
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- From: Angus Rodgers
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