![]() ![]() The set of all neighborhoods is the region of convergence. What this says is that analytic functions, represented by a convergent power series, can have removable singularities (poles) because the power series converges to the value of the rational function in some neighborhood of the pole. A function is analytic if and only if its Taylor series about x0 converges to the function in some neighborhood for every x0 in its domain. Functions of each type are infinitely differentiable, but complex analytic functions exhibit properties that do not hold generally for real analytic functions. There exist both real analytic functions and complex analytic functions, categories that are similar in some ways, but different in others. In mathematics, an analytic function is a function that is locally given by a convergent power series. The following explanation is from wikipedia: Click to expand.In Complex Analysis there is the concept of an "analytic function".
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