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.