My professor made this claim about Taylor Series convergence in my Complex Variables class and I am still not entirely convinced (he said it's explained in the textbook and textbook states,
"we will soon see that it is impossible for the series to converge to $f(z)$ outside this circle"
but they don't actually show that).
Basically the problem I'm having is this:
Let $f(z)$ be expanded in a Taylor series about $z_0$. Let $a$ be the distance from $z_0$ to the nearest singular point of $f(z)$. The claim is that the series converges to $f(z)$ everywhere in the circle $|z-z_0|=a$ and fails to converge to $f(z)$ outside the circle (while nothing can be determined about the points on the circle).
From what I've read in my book, I'm entirely convinced of the first claim that the series converges to $f(z)$ in the circle. It's the other claim that it fails to converge to $f(z)$ outside the circle that is bothering me.
Here is what I know:
I know that the circle for which the Taylor series is everywhere convergent to $f(z)$ is not necessarily the same as the circle throughout which the series converges in general.
I know that $|z-z_0|=a$ is the largest circle within which the series converges to $f(z)$. All this says however is that we can't construct a larger circle for which the series converges to $f(z)$; it still leaves open the possibility that the series could converge to $f(z)$ outside this circle.
I also have this one theorem we proved that states that if a power series centered at $z_0$ converges at the point $z_1$, then the series converges uniformly to an analytic function for all points in the disk $|z-z_0|<|z_1-z_0|$.
Now, the argument my professor made is that suppose that at the point $z=z_1$ the series converged to $f(z)$ where $z_1$ is outside the circle $|z-z_0|=a$. Then by the theorem above, the series would converge to an analytic function inside the circle $|z-z_0|<|z_1-z_0|$. (At this point he stated we had a contradiction and ended our conversation). This would seem like we are onto some kind of contradiction since $f(z)$ is clearly not analytic inside this circle but there is nothing that prevents the series from converging to something other than $f(z)$ outside the circle $|z-z_0|=a$.
I feel like I'm really missing something here or that there is some hole in my understanding. Any help is appreciated!