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Euler's identity (or ``theorem'' or ``formula'') is
![$\displaystyle e^{j\theta} = \cos(\theta) + j\sin(\theta)
$](img91.png)
(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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