12.4 Complex Fourier Series


As we have seen in the preceding two sections, a real function f can be represented by a series of sines and cosines. The functions cos nx, n = 0, 1, 2, … and sin nx, n = 1, 2, … are real-valued functions of a real variable x. The three different real forms of Fourier series given in Definitions 12.2.1 and 12.3.1 will be exceedingly important in Chapters 13 and 14 when we set about to solve linear partial differential equations. However, in certain applications, for example, the analysis of periodic signals in electrical engineering, it is actually more convenient to represent a function f in an infinite series of complex-valued functions of a real variable x such as the exponential functions einx

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