Analytic functions are described by a countable number of coefficients. It is a delusion to think there is an uncountable number of degrees of freedom.
So to make that clear, what we want is some theorem saying: Specifying the value of the analytic function at a countable set of points within the disk, will suffice to specify the function.
Is such a theorem known? I can't think of one off the top of my head. But here, voila, I proved it:
We can make it much easier than that, while proving a stronger theorem. Theorem: a (merely) continuous function from R^n to R^n is determined by countably many numbers. Proof: specify its values at rational points (of which there are only countably many); its values elsewhere are determined by continuity. -- g