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

1 Answer
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''(x) pm f(x) = 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^(ax), we get
a^2 e^(ax) pm e^(ax) = 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.