Imaginary Exponents

GUIDE: Mathematics of the Discrete Fourier Transform (DFT). Imaginary Exponents

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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.

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

\

which is well defined (although we should make sure the series converges for every finite $z$). We have $f(0) \, so the first term is no problem. But what is $f^\? In other words, what is the derivative of $a^x$ at $x=0$? Once we find the successive derivatives of $f(x) \ at $x=0$, we will be done with the definition of $a^z$ for any complex $z$.

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