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        general >> complex analysis >> Schwarz Theorem
        
(Message started by: knightfischer on Aug 22^nd, 2007, 6:10am)

Title: Schwarz Theorem
Post by knightfischer on Aug 22^nd, 2007, 6:10am

In the Schwarz Theorem, it starts out with
f(z) analytical in region |z|<=R, f(0)=0,
then states f(z)/z is analytical in that region.
I do not see how f(z)/z is analytical at z=0. Can anyone help me with this?

Title: Re: Schwarz Theorem
Post by mikedagr8 on Aug 22^nd, 2007, 6:21am

You don't need to post the same topic in different areas of the forum. Someone will respond to you soon enough, As for me I'm of to bed, its's 11.20 here at night, so everywhere else it is probably early morning or evening.

Night All. :-* :P

Title: Re: Schwarz Theorem
Post by Grimbal on Aug 22^nd, 2007, 6:57am

Iceman, you can start one of your riddles now! ::)

Title: Re: Schwarz Theorem
Post by Grimbal on Aug 22^nd, 2007, 7:04am

This is from distant memories, so I offer no warranty.

I think that for z<>0, you can differentiate the function with the usual rule for f/g. For z=0, you can apply L'Hospital rule.

Title: Re: Schwarz Theorem
Post by knightfischer on Aug 22^nd, 2007, 10:27am

I'm not sure L'Hopital applies. Here is my reasoning, given f(z) analytic, and f(0)=0, to show
g(z)=f(z)/z is analytic at z=0, we must show the derivative exists at z=0. That is, the
lim (delta z->0) of {(g(0+delta z)-g(0))/(delta z)}
=lim (delta z ->0) of {(f(delta z)/(delta z) - f(0)/0)/(delta z)}
I can apply L'Hopital to first term in numerator, but there is no limit involved in second term, f(0)/0, so how can I apply L'Hopital to that?

Confused!

Title: Re: Schwarz Theorem
Post by Aryabhatta on Aug 22^nd, 2007, 1:13pm

I am not sure what approach to analytic functions you are using, but using the power series approach we can easily see that

if f(z) = Sum a_i zⁱ is analytic in disc D such that f(z) = 0 (i.e a₀ = 0)

Then H(z) = Sum a_i+1 zⁱ is also analytic in disc D and H(z) = f(z)/z for all z =/= 0. and H(0) = f'(0).

Title: Re: Schwarz Theorem
Post by knightfischer on Aug 23^rd, 2007, 4:13am

Thanks. Your explanation is clear. The book has not yet discussed power series approach to analytic functions. I was trying to see how f(z)/z is analytic, using the definition of an anlytic function as one with a derivative at all points in a region. Using this derivative approach I could not see how it was analytic at z=0.

Title: Re: Schwarz Theorem
Post by Wiliam_smith on Nov 19^th, 2009, 6:43am

Simply because |z| < R . Field.

The same equation we use in Search analytics. Creating a peak within a range of low value keywords. I work for (link removed by moderator)

Title: Re: Schwarz Theorem
Post by SMQ on Nov 19^th, 2009, 7:04am

If you're going to spam the forum by resurrecting two-year-old threads, at least try to make sense. ;)

--SMQ