# nLab hyperbolic sine

## Definitions

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

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

$sinh'' = sinh$

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

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