Proof that Radius of Convergence Extend to Nearest Singularity

Can someone provide a proof for the fact that the radius of convergence of the power series of an analytic function is the distance to the nearest singularity? I've read the identity theorem, but I don't see how it implies that the two functions must be equal everywhere.

Solution 1:

http://en.wikipedia.org/wiki/Analyticity_of_holomorphic_functions

I wrote the initial draft of the article linked above in February 2004, and mostly it's still as I wrote it, although others have contributed.

Postscript:

Let $C$ be a positively oriented circle centered at $a$ that encloses a point $z$ that is closer to $a$ then is any place where $f$ blows up, and that does not enclose, nor pass through, any point where $f$ blows up.

\begin{align}f(z) &{}= {1 \over 2\pi i}\int_C {f(w) \over w-z}\,dw \tag1 \\[10pt] &{}= {1 \over 2\pi i}\int_C {f(w) \over (w-a)-(z-a)} \,dw \tag2 \\[10pt] &{}={1 \over 2\pi i}\int_C {1 \over w-a}\cdot{1 \over 1-{z-a \over w-a}}f(w)\,dw\tag3 \\[10pt] &{}={1 \over 2\pi i}\int_C {1 \over w-a}\cdot{\sum_{n=0}^\infty\left({z-a \over w-a}\right)^n} f(w)\,dw\tag4 \\[10pt] &{}=\sum_{n=0}^\infty{1 \over 2\pi i}\int_C {(z-a)^n \over (w-a)^{n+1}} f(w)\,dw\tag5 \\[10pt] & = \sum_{n=0}^\infty (z-a)^n \underbrace{{1 \over 2\pi i}\int_C {f(w) \over (w-a)^{n+1}} \,dw}_{\text{No $z$ appears here!}}.\tag6 \end{align}

Step $(1)$ above is Cauchy's formula.

Step $(4)$ is summing a geometric series.

Step $(6)$ can be done because "$(z-a)^n$" has no $w$ in it; thus does not change as $w$ goes around the circle $C$.

The fact that no "$z$" appears in the expression in $(6)$ where that is noted, means that the last expression is a power series in $z-a$.

Solution 2:

The answer to this question is a bit hard to find, because there is no general agreement on the name of the relevant theorem. Ahlfors, for example, just calls it « Theorem 3 » in the standard work on Complex Analysis that he wrote (2nd edition, p. 177). The point to remember is that it comes under the heading of « Taylor Series ». You need to take a look at how Cauchy's formula is applied to write down an integral expression for the remainder term of the finite Taylor series, which is expressed as a complex line integral that follows a circle close to the edge of the largest disk where the function f is holomorphic. From the obtained expression for the remainder term, you will see that it approaches zero as the number of terms in the series is increased, which in its turn means that the series is convergent, and that it converges to the value of f at the centre of the circle. And voila! There you have the theorem. Henri Cartan has a very clear statement and proof of the theorem (Théorie élementaire, Paris 1961, p. 73), and you will also find it in Ruel V. Churchill's book.