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

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

This is well defined, provided the series converges for every finite $z$ (see Problem 9). 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 $z$.