# nLab hyperbolic cosine

## Definitions

The hyperbolic cosine function is the function $\cosh \;\colon\; \mathbb{R} \to \mathbb{R}$ from the real numbers to themselves which is characterized by the following condition:

1. $\cosh$ is the unique solution among smooth functions to the differential equation/initial value problem

$cosh '' = cosh$

(where a prime indicates the derivative) subject to the initial conditions

\begin{aligned} cosh(0) &= 1 \\ cosh'(0) & = 0 \,. \end{aligned}