Euler's Identity

Euler's identity (or ``theorem'' or ``formula'') is

$$e^{j\theta} = \cos(\theta) + j\sin(\theta)$$   (Euler's Identity)

To ``prove'' this, we will first define what we mean by `` $e^{j\theta }$''. (The right-hand side, $\cos(\theta) + j\sin(\theta)$, is assumed to be understood.) Since $e$ is just a particular real 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.