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The Princeton Companion to Mathematics by Imre Leader, June Barrow-Green, Timothy Gowers

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III.79 Riemann Surfaces

Alan F. Beardon

Let D be a region (that is, a connected open set) in the complex plane. If f is a complex-valued function defined on D, then we can define its derivative just as we would for real-valued functions defined on subsets of Image: the derivative of f at w is the limit as z tends to w of the “difference quotient” (f(z) - f(w)) / (z - w). Of course, this limit does not necessarily exist, but if it exists for every w in D, then f is said to be analytic, or holomorphic, on D. Analytic functions have amazing properties; for example, if a function is analytic in a region, then it automatically has a Taylor-series expansion ...

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