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 section 9.3 
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Joined: 07 Jun 2008, 08:21
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Post section 9.3
I understand section 9.2 very well.

However, I have been reading and re-reading section 9.3 for a while now and find that I do not understand it.

On page 163, and particularly in the text under Figure 9.7, how can Penrose talk as though all the negative frequency components in the z-plane are within the unit circle and all the positive ones outside the unit circle when in section 9.2 and in Figure 9.5 he is showing that some negative frequency components are also outside the unit circle?

Similarly for the positive frequency components.
I understand the transformation from z to t perfectly, having done exercises 8.7 and 9.5.

Any ideas, explanations anyone?


14 Apr 2010, 07:36

Joined: 29 Jan 2010, 19:21
Posts: 11
Post Re: section 9.3
Vasco, my understanding is that on section 9.2 Penrose is talking about any z. If B<|z|<A the series will converge.
In section 9.3 he is talking about an z where |z|=1 .
In 9.2 (pg 160) he says: "Does this mean that, for convergence of the Fourier series, we necessarily require the unit circle to lie within the annulus of convergence?
This would indeed be the case (...) "
And then in section 9.3 he is talking about a (any?) series that will converge because |z|=1. My understanding is that we are not worried anymore about what happens inside/outside the unit circle, we are just worried about the unit circle itself ...
Does this make sense for you?

Rodrigo


14 Apr 2010, 14:20
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Post Re: section 9.3
Rodrigo
Thanks for your reply. But if he is only concerned with |z|=1 in section 9.3 why does he then say, on page 163, just underneath the transformation formula:

"The interior of the unit circle in the z-plane corresponds to the upper half of the t-plane and the exterior of the unit circle corresponds to the lower half of the t-plane. Hence, positive-frequency functions of t are those that extend holomorphically into the lower half of t and negative-frequency ones, into the upper half-plane."

I understand that the transformation into the t-plane transforms the unit circle in the z-plane to the real axis as shown in Figure 9.7, but in the quotation above he is talking about the inside and outside of the unit circle in the z-plane and I understand how the transformation maps these into the upper and lower halves of the t-plane.

What I don't understand is the "Hence...." part. He seems to be saying that positive frequency functions are confined to the outside of the unit circle and vice versa, and I don't see this. It seems to me that there are positive and negative frequency functions defined over the whole z-plane.


14 Apr 2010, 14:47

Joined: 29 Jan 2010, 19:21
Posts: 11
Post Re: section 9.3
As far as I understand he is basically talking about the Riemann sphere.
Figure 8.8 (pag 145) shows exactly the same relation (upper-half plane t mapped to the interior unit-circle.
I think that what is confusing you is that the positive-frequence is defined as F^+ = \alpha_r z^{-r} and negative-frequence is F^- = \alpha_r z^{r} (look at the beggining of page 160).
So figure 8.8 shows the negative-frequence mapping (because we talking about z) and for positive-frequence we must think about w=\frac{1}{z} ...
Rodrigo


14 Apr 2010, 15:45
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Post Re: section 9.3
No, I'm not confused between the functions F^+ and F^-, and I understand Fig 8.8.
The point is that if you choose a z with|z|<1, then w is outside the unit circle and vice versa, so z can be inside the unit circle and 1/z outside and vice versa.


14 Apr 2010, 16:00

Joined: 29 Jan 2010, 19:21
Posts: 11
Post Re: section 9.3
Yes. I think you are right. It is just a matter of convention.
But you will never have |z|>1 and |w|>1 at the same time, since, by definition w=\frac{1}{z}, so, if one maps inside, the other maps outside.
Rodrigo


14 Apr 2010, 20:57
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