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 may be
generalized to a function of a complex variable by simply
substituting the complex variable for the real variable
in the Taylor series expansion of .

Let
, where is any positive real number and is
real. The Taylor
series expansion about (``Maclaurin series''),
generalized to the complex case is then

(3.1)

This is well defined, provided the series converges for every
finite (see Problem 9). We have
, so the first term is no problem. But what is
? In
other words, what is the derivative of at ? Once we find
the successive derivatives of
at , we will have
the definition of for any complex .