acosh() [9/11]

var stan::math::acosh ( const var &  x )

inline

The inverse hyperbolic cosine function for variables (C99).

For non-variable function, see acosh().

The derivative is defined by

\(\frac{d}{dx} \mbox{acosh}(x) = \frac{x}{x^2 - 1}\).

\[ \mbox{acosh}(x) = \begin{cases} \textrm{NaN} & \mbox{if } x < 1 \\ \cosh^{-1}(x) & \mbox{if } x \geq 1 \\[6pt] \textrm{NaN} & \mbox{if } x = \textrm{NaN} \end{cases} \]

\[ \frac{\partial\, \mbox{acosh}(x)}{\partial x} = \begin{cases} \textrm{NaN} & \mbox{if } x < 1 \\ \frac{\partial\, \cosh^{-1}(x)}{\partial x} & \mbox{if } x \geq 1 \\[6pt] \textrm{NaN} & \mbox{if } x = \textrm{NaN} \end{cases} \]

\[ \cosh^{-1}(x)=\ln\left(x+\sqrt{x^2-1}\right) \]

\[ \frac{\partial \, \cosh^{-1}(x)}{\partial x} = \frac{1}{\sqrt{x^2-1}} \]

Parameters

x	The variable.

Returns

Inverse hyperbolic cosine of the variable.

Definition at line 63 of file acosh.hpp.