Euler's Theorem

GUIDE: Mathematics of the Discrete Fourier Transform (DFT). Euler's Theorem

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Euler’s Theorem

Euler’s Theorem (or “identity” or “formula”) is

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To “prove” this, we must first define what we mean by “$e^{j\.” (The right-hand side is assumed to be understood.) Since $e$ is just a particular number, we only really have to explain what we mean by imaginary exponents. (We’ll also see where $e$ 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 $a^x$, where$x$ is any real number, and $a>0$ is any positive real number.



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