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
![$ z$](http://www.dsprelated.com/josimages_new/mdft/img30.png)
![$ f(0) \isdeftext a^0
= 1$](http://www.dsprelated.com/josimages_new/mdft/img279.png)
![$ f^\prime(0)$](http://www.dsprelated.com/josimages_new/mdft/img280.png)
![$ a^x$](http://www.dsprelated.com/josimages_new/mdft/img230.png)
![$ x=0$](http://www.dsprelated.com/josimages_new/mdft/img144.png)
![$ f(x) \isdeftext a^x$](http://www.dsprelated.com/josimages_new/mdft/img281.png)
![$ x=0$](http://www.dsprelated.com/josimages_new/mdft/img144.png)
![$ a^z$](http://www.dsprelated.com/josimages_new/mdft/img282.png)
![$ z$](http://www.dsprelated.com/josimages_new/mdft/img30.png)
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Derivatives of f(x)=a^x
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A First Look at Taylor Series