Euler's identity (or ``theorem'' or ``formula'') is

(Euler's Identity)
To ``prove'' this, we will first define what we mean by
``

''. (The right-hand side,

, is assumed to be understood.) Since

is just a
particular
real number, we only really have to explain what we mean by
imaginary exponents. (We'll also see where

comes from in the
process.) Imaginary exponents will be obtained as a generalization of
real exponents. Therefore, our first task is to define exactly what
we mean by

, where

is any real number, and

is any
positive real number.
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