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Adapted from wikipedia:
A holomorphic function is a complex-valued function of one or more complex variables (inputs that are complex numbers) that is complex differentiable in a neighborhood of every point in its domain. Loosely speaking, it is "smooth" everywhere.
The existence of a complex derivative in a neighborhood is a very strong condition, for it implies that any holomorphic function is actually infinitely differentiable and equal to its own Taylor series.
The term "analytic function" is often used interchangeably with "holomorphic function", although the word "analytic" is also used in a broader sense to describe any function (real, complex, or of more general type) that can be written as a convergent power series in a neighborhood of each point in its domain. The fact that the class of complex analytic functions coincides with the class of holomorphic functions is a major theorem in complex analysis.
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