The sinh function is a hyperbolic function. It is also known as the *hyperbolic sine* function.

## Equation and graph

The sinh function is defined as:

$$
\sinh{x} = \frac{e^{x}-e^{-x}}{2}
$$

Here is a graph of the function:

## sinh as average of two exponentials

The sinh function can be interpreted as the average of two functions, $e^{x}$ and $-e^{-x}$. This animation illustrates this:

## Other forms of the equation

If we multiply the top and bottom of the original equation for the sinh function by $e^{x}$ we get:

$$
\sinh{x} = \frac{e^{2x}-1}{2e^{x}}
$$

Alternatively, if we multiply the top and bottom of the original equation for the sinh function by $e^{-x}$ we get:

$$
\sinh{x} = \frac{1-e^{-2x}}{2e^{-x}}
$$

These alternative forms are sometimes useful.