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.
Let , where is any positive real number. The Taylor series expansion expansion about (“Maclaurin series”), generalized to the complex case is then
which is well defined (although we should make sure the series converges for every finite ). 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 be done with the definition of for any complex .