Re: [Fwd: Two questions in Complex Analysis]

From: Roger L. Bagula (rlbtftn_at_netscape.net)
Date: 07/21/04 

Date: Wed, 21 Jul 2004 22:37:20 GMT

Basic questions of complex analysis are
very important in understanding most fractals of the Julia/ Mandelbrot type.
There is a clear transition from complex analysis to
complex dynamics ( one needs to know the first
to understand the second).
I think the notation and questions are both interesting to
sci.fractals and sci.nonlinear. Actually professors who teach such
courses are usually taken from those who do
complex dynamics or Hardy space types.
Roger L. Bagula wrote:
>
>
> -------- Original Message --------
> Subject: Two questions in Complex Analysis
> Date: 21 Jul 2004 10:54:47 -0700
> From: eytan@tradertools.com (Mimsy Boro)
> Organization: http://groups.google.com
> Newsgroups: sci.math
>
> I have to short question in Complex Analysis which are part of an
> excersize in my course. I'm quite stuck, any hint would be helpful.
>
> 1. Let f(z) = \sum_{n=0}^{\infty}a_{n}z^n be an analytic function in
> an open circle of radius R around 0. It is known that | f(z) | <= M.
> Let z_{0} be the closest 0 of f(z). Prove that:
> |z_{0}| >= R|a_{0}|/(M+|a_{0}|)
>
> 2. Let g(z) be an analytic function that is defined for |z|>1.
> It is known that Im ( f(z) ) = 0 for each z in [1,\infty}.
> Prove that Im( f(z) ) = 0 for each z in (-\infty,1].
>
> Thanks In Advance,
> 

-- 
Respectfully, Roger L. Bagula
tftn@earthlink.net, 11759Waterhill Road, Lakeside,Ca 92040-2905,tel: 
619-5610814 :
URL :  http://home.earthlink.net/~tftn
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