naginterfaces.library.specfun.arccosh

naginterfaces.library.specfun.arccosh(x)

arccosh returns the value of the inverse hyperbolic cosine, $arccosh\left(x\right)$. The result is in the principal positive branch.

For full information please refer to the NAG Library document for s11ac

https://support.nag.com/numeric/nl/nagdoc_30.3/flhtml/s/s11acf.html

Parameters

xfloat

The argument $x$ of the function.

Returns

achxfloat

The value of the inverse hyperbolic cosine, $arccosh\left(x\right)$.

Raises

NagValueError

(errno $1$)

On entry, $x=⟨value⟩$.

Constraint: $x\ge 1.0$.

The function has been called with an argument less than $1.0$, for which $arccosh\left(x\right)$ is not defined.

Notes

arccosh calculates an approximate value for the inverse hyperbolic cosine, $arccosh\left(x\right)$. It is based on the relation

$$arccosh\left(x\right)=ln(x+\sqrt{x^2-1})\text{.}$$

This form is used directly for $1<x<{10}^k$, where $k=n/2+1$, and the machine uses approximately $n$ decimal place arithmetic.

For $x\ge {10}^k$, $\sqrt{x^2-1}$ is equal to $\sqrt{x}$ to within the accuracy of the machine and hence we can guard against premature overflow and, without loss of accuracy, calculate

$$arccosh\left(x\right)=ln\left(2\right)+ln\left(x\right)\text{.}$$

References

NIST Digital Library of Mathematical Functions