The exponential function (exp)
The exponential function \(e^x\), sometimes written as \(\mathrm{exp}(x)\), is a famous function with many applications, far more than just exponential growth and decay. It is defined by the power series 1 2,
\begin{equation*}
\begin{array}{lcl}
\displaystyle e^x &=& \displaystyle \sum_{n=0}^\infty \frac{x^n}{n!} \\
&=& \displaystyle 1 + x + \frac{x^2}{2} + \frac{x^3}{6} + \frac{x^4}{24} + \frac{x^5}{120}+\cdots
\end{array}
\end{equation*}
with \(n!\) the faculty of the integer \(n\) and the number \(e\,=\,\sum_{n=0}^\infty \frac{1}{n!}\,\approx\,2.718....\) the constant of Euler, after the Swiss mathematician Leonhard Euler (1707 - 1783) 3.