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## Euler’s Theorem

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

To “prove” this, we must first define what we mean by “.” (The right-hand side is assumed to be understood.) Since is just a particular 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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