# Why are the hyperbolic functions defined using e? Why not some other constant?

May 25, 2018

$e$ is the base of the natural logarithm and comes out in many differential equations.

#### Explanation:

It's primarily because $e$ has a very nice property of its derivative being really clean.

All of the trig functions are defined by the differential equation
$f ' ' \left(x\right) \pm f \left(x\right) = 0$
where the minus is hyperbolic and the plus is circular.

There are two solutions for each, both exponentials. Assuming the functions have form ${e}^{a x}$, we get
${a}^{2} {e}^{a x} \pm {e}^{a x} = 0 \implies {a}^{2} \pm 1 = 0$

so for the circular case, we get $a = \pm i$ and for the hyperbolic case, we get $a = \pm 1$. In either case, we get these exponentials and their sums as the solutions of the differential equation.