Abstract
Abstract We shall assume that readers know the basic properties of the real trigonometric, exponential, and hyperbolic functions, including the formulae for their derivatives from which the Maclaurin expansions of these functions are obtained. For most, this knowledge will be founded on a naive treatment of these functions, relying on elementary geometry and trigonometry. A few may have seen the naive approach superseded by a more analytical one, in which functions are defined by power series and their expected properties are then derived from the properties of these series. In the complex case, a geometric approach to the elementary functions is no longer available, but this is not a problem. Power series definitions serve admirably, since convergent complex power series behave so well. We begin by investigating that most fundamental of functions, the exponential function.
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