Vasco, in your third solution to exercise 9.3 you consider a region inside the unit circle of the z plane. but surely if z=e^(iwx) then z only takes values on the unit circle itself.
I think your solution (b) is the correct one

I don't agree.
Penrose says, in the statement of the exercise, that F(z) is analytic on the unit circle. This means that it must be equal to its Taylor series in the neighbourhood of each point on the unit circle, which means that it is analytic/holomorphic in an annulus which includes the unit circle.
The variable z only lies on the unit circle when we wish to recover the Fourier series for f(x). Anywhere inside the annulus there is a Laurent series for F(z) which coincides with the Fourier series when z lies on the unit circle.

But then again maybe you're right. I'm a bit confused now. I'll give it some more thought. Thanks for the feedback.

Hi Tony,
I have thought some more about what you say and I now agree with you that my (b) solution is the best answer to exercise 9.2., since it just deals with the case when the function is defined on the unit circle.

Solution (c) is a generalisation of (b) which covers the cases where the function is defined over an annulus and is not required for the proof of exercise 9.2.

Vasco
Edited on 23rd August 2010 at 7:10 BST