I think I am supposed to use Rouche Theorem and Cauchy Argument Principle for this problem as well:

http://i3.photobucket.com/albums/y57...ty/prob2-1.jpg

Any ideas? Any help is appreciated! thanks

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- Dec 3rd 2007, 09:06 PMhanahoushow that f has a fixed point
I think I am supposed to use Rouche Theorem and Cauchy Argument Principle for this problem as well:

http://i3.photobucket.com/albums/y57...ty/prob2-1.jpg

Any ideas? Any help is appreciated! thanks - Dec 4th 2007, 07:41 AMThePerfectHacker
Define $\displaystyle g(z) = f(z) - z$ and use Rouche's theorem by showing that $\displaystyle g(z)$ has a zero in the disk. That will mean there is $\displaystyle a\in D(0,1)$ such that $\displaystyle g(a)=0\implies f(a) = a$ and hence $\displaystyle f$ has a fixed point.