The Exponential Function

$$\begin{eqnarray*} \mrr {e^x}{\isdef }{\displaystyle\lim_{n\to\infty}\left(1+\frac{x}{n}\right)^n}% {e^x}{=}{\displaystyle\sum_{n=0}^\infty\frac{x^n}{n!}} \mr {e^{j\theta}}{\cos(\theta)+j\sin(\theta)}% {e^{jn\theta}}{\cos(n\theta)+j\sin(n\theta)} \\ [10pt] \mr {\sin(\theta)}{\frac{e^{j\theta}-e^{-j\theta}}{2j}}% {\cos(\theta)}{\frac{e^{j\theta}+e^{-j\theta}}{2}} \mrr {e}{=}{2.7\,1828\,1828\,4590\,\ldots}{}{}{} \end{eqnarray*}$$