## Definitions

### General

The hyperbolic sine function is defined as:

\[ \sinh{x} = \frac{\mathrm{e}^x-\mathrm{e}^{-x}}{2} \]

The graph of the hyperbolic sine function is shown in the figure below. It is a monotonic function unlike the trigonometric sine, which is periodic .

The points \( (\cosh{t}, \sinh{t})\) form the right wing of an equilateral hyperbola (see figure below), just like the trigonometric cosine, sine pairs form a circle. Parameter t is the half area between the hyperbola, the x-axis and a ray from origin to the \( (\cosh{t}, \sinh{t})\) point.

### Series

All hyperbolic functions can be defined in an infinite series form. Hyperbolic sine function can be written as:

\[ \begin{split} \sinh x & = \sum_{n=0}^{\infty}\frac{x^{2n+1}}{(2n+1)!} = \\ & = x + \frac{x^3}{6} + \frac{x^5}{120} + \frac{x^7}{5040} \cdots \end{split} \]

From the expanded form of the series it can be seen that the higher terms become insignificant, for values of x close to zero, resulting in the following quite useful approximation:

\[ \sinh x \approx x, \quad x \to 0 \]

### Properties

The derivative of the hyperbolic sine function is the hyperbolic cosine:

\[ \left(\sinh{x}\right)' = \cosh{x} \]

The integral of the hyperbolic sine is given by:

\[ \int \sinh{x} \, \mathrm{d}x = \cosh{x} +C \]

### Identities

\[ \begin{split} & \sinh{\left(-x\right)} & = -\sinh{x} \\ \\ & \sinh{\left(2 x\right)} & = 2\sinh{x}\cosh{x} \\ \\ & \sinh{\left(x + y\right)} & = \sinh{x}\cosh{y} + \cosh{x}\sinh{y} \\ \\ & \sinh{x} + \sinh{y} & = 2\sinh{\frac{x+y}{2}} \cosh{\frac{x-y}{2}} \\ \\ & \sinh{\left(\frac{x}{2}\right)} & = \frac{\sinh{x}}{\sqrt{2\cosh{x}+2}} \\ \\ & \sinh{x} + \cosh{x} & = \mathrm{e}^{x} \\ \\ & \sinh{x} - \cosh{x} & = -\mathrm{e}^{-x} \\ \\ & \cosh^2{x} - \sinh^2{x} & = 1 \\ \\ \end{split} \]