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:
Comparing the Maclaurin expansion for with that of and proves Euler's identity. Recall from introductory calculus 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.