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        general >> complex analysis >> Cplx Problem
        
(Message started by: axterix on Jul 22^nd, 2005, 4:15pm)

Title: Cplx Problem
Post by axterix on Jul 22^nd, 2005, 4:15pm

Find all function f analytic in the disk D such that |z| < R which satisfies f(0) = i and |f(z)| <=1 or all z in D, and where R is any number greater than zero.

Title: Re: Cplx Problem
Post by Michael_Dagg on Jul 23^rd, 2005, 9:01am

This problem slightly misstated. There is only one such function and it is f(z) = i.

Title: Re: Cplx Problem
Post by axterix on Jul 26^th, 2005, 9:21pm

That is the right answer but what part is wrong?

Title: Re: Cplx Problem
Post by axterix on Jul 26^th, 2005, 9:36pm

Ah, never mind - I see it.

How do you know or how did show that there is only one function f(z) = i that satisfies the properties?

Title: Re: Cplx Problem
Post by Eigenray on Jul 27^th, 2005, 10:56am

Read up on the maximum modulus principle. Or the open mapping theorem.

Title: Re: Cplx Problem
Post by Michael_Dagg on Jul 27^th, 2005, 12:04pm

The problem is a frequent exercise in CA books and is generally associated with the first hint Eigenray gave.