Back to e^(j theta)

GUIDE: Mathematics of the Discrete Fourier Transform (DFT). Back to e^(j theta)

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Back to e^(j theta)

We’ve now defined $a^z$ for any positive real number $a$ and any complex number $z$. Setting $a=e$ and $z=j\ gives us the special case we need for Euler’s identity. Since $e^z$ is its own derivative, the Taylor series expansion for for $f(x)=e^x$ is the simplest series there could be:

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The simplicity comes about because $f^{(n)}(0)=1$ for all $n$ and because we chose to expand about the point $x=0$. We of course define

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Note that all even order terms are real while all odd order terms are imaginary. Separating out the real and imaginary parts gives

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Comparing the Maclaurin expansion for $e^{j\ with that of$\ and $\ proves Euler’s identity. Recall that

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so that

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Plugging into the general Maclaurin series gives

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Separating the Maclaurin expansion for $e^{j\ into its even and odd terms (real and imaginary parts) gives

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thus proving Euler’s identity.

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