## Definitions

### General

The hyperbolic cosine function is defined as:

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

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

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 cosine function can be written as:

\[ \begin{split} \cosh x & = \sum_{n=0}^{\infty}\frac{x^{2n}}{(2n)!} = \\ & = 1 + \frac{x^2}{2} + \frac{x^4}{24} + \frac{x^6}{720} \cdots \end{split} \]

### Properties

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

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

The integral of the hyperbolic cosine is given by:

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

### Identities

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