## 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 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

This is well defined, provided the series

*converges*for every finite (see Problem 8). 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 .

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Derivatives of f(x)=a^x

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A First Look at Taylor Series