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 : 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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