Re: Section 7.4 of RTR on Analytic Continuation

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.