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: use monte carlo integration to compute the maclaurin series coefficients. With probability=1, any particular series coefficient will get computed in this way with error that in the infinite-points limit goes to zero. That proves that a countable set of points must exist, that works to compute any finite-numbered coefficient exactly; and hence, theorem proven.
I don't follow your proof, but it must have a flaw, because the theorem is false. The exponential function and the constant function 1 agree at 2k * pi * i, a countable number of points. Choose some simply connected region containing all of these, and use the Riemann mapping theorem to get two analytic functions on the (open) disk that agree at countably many points. One of these functions will not extend to an analytic function on the closed disk, because the actual theorem is that if two complex analytic functions agree on a sequence of points that has a limit, and at that limit point, and are analytic at that limit point, they are identical. I don't remember the proof, but I think it was covered in my intro complex analysis course, so it doesn't use a lot of machinery. Andy Latto andy.latto@pobox.com
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