Page 131 of my paperback RTR. I think the paragraph under Fig 7.5 doesn't make sense. It refers to section 4.4 and the series it quotes is for but I think it should be and may be even . Maybe there are other mistakes.
Can anyone suggest a re-writing that makes sense? Or am I wrong?

Vasco,
The result that Prof. Penrose quotes is absolutely correct.To get some familiarity with the concept I suggest you sum the given series using properties of a geometric progression.

Now this expansion is valid for less than unity.However,if you observe the RHS carefully you will see that the function is continuous at every point except at and .Therefore,the 'smoothness' of the function can fail at only two points.

On the other hand,the expansion was initially summed with the unit circle as the circle of convergence.We had no clue whether the series converges outside its circle.The above result,however points towards the fact that the series may converge outside its circle except at i and -i.

Thanks very much. You have cleared that up nicely for me.