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Understanding exercise 9.3
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Joined: 07 Jun 2008, 08:21
Posts: 235
Understanding exercise 9.3
I think that I am missing something important when I read section 9.2 and 9.3 of RTR.

On page 157 at the end of section 9.1 we have the Laurent series for F(z) and the result of exercise 9.2 at the foot of this page gives us an expression for the coefficients in this Laurent series.

The series is valid in an annulus containing the unit circle as explained in section 9.2.
It seems to me that this Laurent series is unique, since we know all the coefficients.

Penrose then expresses the Laurent series as the sum of two functions and a constant, by expressing the sum of the terms of positive powers of z as one function, and the sum of the terms of negative powers of z as another function.

The power series in positive powers of z (F say) converges inside a disk surrounding the origin and the power series in 1/z (G say) converges outside a circle surrounding the origin.

If these two regions overlap then in the overlap (annulus), the Laurent series converges.

At the beginning of section 9.3 Penrose sets an exercise 9.3, which asks us to show why the holomorphic extension of F or G into the two hemispheres of the Riemann sphere guarantees the uniqueness of the split into F and G.

It seems to me that although F + G is only defined on the annulus, F is defined in the disc bounded by the outer circle of the annulus, and G is defined from the inside circle of the annulus out to infinity. So this means that F & G are also defined on the Riemann sphere as holomorphic functions.
See diagram 9.5 on page 159 for a picture of all this.

So what is this 9.3 exercise all about, since it seems to me that F and G are already holomorphic on the Riemann sphere?

01 Jun 2010, 08:23

Joined: 26 Mar 2010, 04:39
Posts: 109
Re: Understanding exercise 9.3
vasco wrote:
So what is this 9.3 exercise all about, since it seems to me that F and G are already holomorphic on the Riemann sphere?

Not quite. As you said yourself, in general F is only holomorphic in a neighbourhood of the origin and G is only holomorphic in a neighbourhood of complex infinity, so neither need be holomorphic on the entire sphere. If these neighbourhoods overlap then there is an annulus where both functions are simultaneously holomorphic and thus where F+G is defined.

The statement of 9.3 can be elaborated as follows. Let F be any function defined on the unit circle. Show that there exist functions F_+, F_- defined uniquely up to a constant term, such that F = F_+ + F_- and where F_+ is holomorphic in a neighbourhood of complex infinity and F_- is holomorphic in a neighbourhood of the origin.

Does this make sense?

01 Jun 2010, 11:36
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Joined: 07 Jun 2008, 08:21
Posts: 235
Re: Understanding exercise 9.3
Robin
Thanks for your reply. After reading your reply I don't think I can have explained myself well enough. I will produce a more detailed explanation using Latex and attach it as a pdf file. I would be grateful if you could find the time to have a look at it.
Vasco

01 Jun 2010, 17:32
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