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