e^(j theta)
We've now defined

for any positive
real number 
and any
complex number 
. Setting

and

gives us the
special case we need for
Euler's identity. Since

is its own
derivative, the
Taylor series expansion for

is one of
the simplest imaginable infinite series:

The simplicity comes about because

for all

and because
we chose to expand about the point

. We of course define
Note that all even order terms are real while all odd order terms are
imaginary. Separating out the real and imaginary parts gives
Comparing the Maclaurin expansion for

with that of

and

proves Euler's identity. Recall
from introductory
calculus that
so that
Plugging into the general
Maclaurin series gives
Separating the Maclaurin expansion for

into its even and odd
terms (real and imaginary parts) gives
thus proving Euler's identity.
Next Section: Back to Mth RootsPrevious Section: Back to e