## Euler's Identity

*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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Positive Integer Exponents

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