##
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.

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