Please read the attached and tell me if you agree this is an error in the book.
Thanks

I now realise that due to a couple of thing's Penrose writes in chapter 9, I ended up confused and as a result I published the post above.
I now realise that as long as you understand that when Penrose writes under Figure 9.5(b):

"...where refers to the extended northerm hemisphere and to the extended southern hemisphere.",

that the word "extended" refers to the parts which make up the annulus of convergence;

whereas in section 9.3, on page 161, in paragraph one, the words:

"...extends holomorphically into the southern hemisphere...extending holomorphically into the northern hemisphere...",

refer to the extending of the functions and defined on the unit circle into the south and north hemispheres respectively, then you will not get confused.

It was these two different uses and meanings attached to the words extended/extends/extending that lead to my confusion.

So to sum up I am withdrawing what I wrote in the pdf file attached to the above post.

Remember that converges if , and this means the whole northern hemisphere and the southern hemisphere above , the opposite for .
The annulus is where both and converges. But if you are outside the annulus you can find that or converge.
Rodrigo