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Imaginary Exponents

We may define imaginary exponents the same way that all sufficiently smooth real-valued functions of a real variable are generalized to the complex case--using Taylor series. A Taylor series expansion is just a polynomial (possibly of infinitely high order), and polynomials involve only addition, multiplication, and division. Since these elementary operations are also defined for complex numbers, any smooth function of a real variable $ f(x)$ may be generalized to a function of a complex variable $ f(z)$ by simply substituting the complex variable $ z = x + jy$ for the real variable $ x$ in the Taylor series expansion of $ f(x)$ .

Let $ f(x) \isdef a^x$ , where $ a$ is any positive real number and $ x$ is real. The Taylor series expansion about $ x_0=0$ (``Maclaurin series''), generalized to the complex case is then

$\displaystyle a^z \isdef f(0)+f^\prime(0)(z) + \frac{f^{\prime\prime}(0)}{2}z^2 + \frac{f^{\prime\prime\prime}(0)}{3!}z^3 + \cdots\,. \protect$ (3.1)

This is well defined, provided the series converges for every finite $ z$ (see Problem 8). We have $ f(0) \isdeftext a^0
= 1$ , so the first term is no problem. But what is $ f^\prime(0)$ ? In other words, what is the derivative of $ a^x$ at $ x=0$ ? Once we find the successive derivatives of $ f(x) \isdeftext a^x$ at $ x=0$ , we will have the definition of $ a^z$ for any complex $ z$ .

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``Mathematics of the Discrete Fourier Transform (DFT), with Audio Applications --- Second Edition'', by Julius O. Smith III, W3K Publishing, 2007, ISBN 978-0-9745607-4-8
Copyright © 2024-04-02 by Julius O. Smith III
Center for Computer Research in Music and Acoustics (CCRMA),   Stanford University