Definition of hyperbolic trig functions

Question

I was doing some homework for my complex analysis class and ran into a personal question. I haven't worked a lot with hyperbolic trig functions (e.g. $\sinh (x)$, $\cosh(x)$, etc.) so this question may be trivial. However, I was wondering if there is an alternative definition of the hyperbolic trig functions (namely, $\sinh(x)$ and $\cosh(x)$ as the other functions can be generated from these two) other than the usual exponential forms. Furthermore, is there a different definition for $x \in \mathbb{R}$ than $x \in \mathbb{C}$? Just to be clear, I am not asking about the geometric interpretation of these functions, just the analytic definitions.

Answer 1

The usual sine and cosine are the solutions to the differential equation $f''(x)=-f(x)$ that satisfy $\sin(0)=0$, $\sin'(0)=1$ and $\cos(0)=1$, $\cos'(0)=0$.

The hyperbolic functions are solutions to $f''(x)=f(x)$ with the same boundary conditions.

These characterizations work with either $\mathbb R$ or $\mathbb C$ as the domain.