From: ullrich@math.okstate.edu (David C. Ullrich)
Subject: Re: Analytic functions
Date: Sat, 07 Jul 2001 14:54:41 GMT
Newsgroups: sci.math
Summary: Small agreements between analytic functions imply equality

On Sat, 7 Jul 2001 17:24:09 +0300, "Richard Burt"
<shakespeare@fastmail.ca> wrote:

>"... are analytic functions and these have a remarkable property:
>they need only to be known along a suitable line or curve in the
>complex plane in order to be known everywhere, for all complex
>arguments."
>
>Does this have to do something with Cauchy-Riemann conditions?
>Or what?

Has more to do with the fact that if f is analytic (in a connected
set) and not identically zero then the zeroes of f are isolated
(if f=g on a curve then f-g=0 on that curve and hence the
zeroes of f-g are not isolated; hence f-g muct be identically
zero.)

The fact that the zeroes are isolated has to do with the
fact that an analytic function is given by a power series:
Suppose for example that f(0) = 0 but f does not vanish
identically near the origin. Then f has a power series
expansion

f(z) = a_n x^n + a_(n+1) x^(n+1) + ...

valid near the origin, where n > 0 and a_n <> 0. Now
you can show that if z <> 0 is small enough then
the first term is larger than the sum of all the
other terms, and hence if z <> 0 is close enough
to 0 you must have f(z) <> 0.

>Richie
>
>
>


David C. Ullrich
**********************************
Disclaimer: No, I don't know that Stephen Norris has ever stated 
explicitly that log(2) = 1. It's an immediate consequence of 
things he's read somewhere in Hardy&Wright.