Globally analytic function

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Definition

Consider a function f defined on all of \R. We say that f is globally analytic if it satisfies the following equivalent conditions:

  1. There is a point x_0 \in \R such that the Taylor series of f at x_0 exists and converges to f on all of \R.
  2. There is a point x_0 \in \R such that f has a power series at x_0 that converges to f on all of \R.
  3. For every point x_0 \in \R, the Taylor series of f at x_0 exists and converges to f on all of \R.
  4. For every point x_0 \in \R, f has a power series at x_0 that converges to f on all of \R.