# Inverse hyperbolic functions

## Definition of Inverse Hyperbolic Functions

**Inverse hyperbolic functions**are the inverses of the hyperbolic functions sinh, cosh, and tanh, and are often denoted as arcsinh, arccosh, and arctanh respectively.**Arcsinh(x)**is the function defined by the equation`y = arcsinh(x)`

if and only if`x = sinh(y)`

.- Similarly,
**arccosh(x)**and**arctanh(x)**are respectively defined by the equations`y = arccosh(x)`

if and only if`x = cosh(y)`

, and`y = arctanh(x)`

if and only if`x = tanh(y)`

. - Like hyperbolic function,
**inverse hyperbolic functions**are also functions of real variables.

## Properties of Inverse Hyperbolic Functions

- Unlike their hyperbolic counterparts,
**inverse hyperbolic functions**do not possess symmetry properties because they are neither odd nor even. **Inverse hyperbolic functions**can be expressed in terms of logarithms:`arcsinh(x) = ln(x + sqrt(1 + x^2))`

`arccosh(x) = ln(x + sqrt(x^2 - 1))`

{only defined for x≥1}-
`arctanh(x) = 0.5 * ln((1 + x) / (1 - x))`

{only defined forx <1}

- The function
**arcsinh(x)**is an increasing function, while**arccosh(x)**is a decreasing function. - The composition of a function and its inverse is the identity function, for example
`cosh(arcsinh(x)) = sqrt(1 + x^2)`

and`sinh(arccosh(x)) = sqrt(x^2 - 1)`

## Calculating with Inverse Hyperbolic Functions

- Inverse hyperbolic functions, like regular hyperbolic functions, can be manipulated using standard rules of algebra and calculus.
- The
**derivatives**of the inverse hyperbolic functions can be derived as follows:`(d/dx) arcsinh(x) = 1 / sqrt(1 + x^2)`

`(d/dx) arccosh(x) = 1 / sqrt(x^2 - 1)`

{only defined for x>1}-
`(d/dx) arctanh(x) = 1 / (1 - x^2)`

{only defined forx <1}

- The
**integrals**of the inverse hyperbolic functions can be derived using integration by parts or other common methods.