naginterfaces.library.specfun.arcsinh

naginterfaces.library.specfun.arcsinh(x)

arcsinh returns the value of the inverse hyperbolic sine, $arcsinh\left(x\right)$.

For full information please refer to the NAG Library document for s11ab

https://support.nag.com/numeric/nl/nagdoc_30.2/flhtml/s/s11abf.html

Parameters

xfloat

The argument $x$ of the function.

Returns

ashxfloat

The value of the inverse hyperbolic sine, $arcsinh\left(x\right)$.

Notes

arcsinh calculates an approximate value for the inverse hyperbolic sine of its argument, $arcsinh\left(x\right)$.

For $\left|x\right|\le 1$ it is based on the Chebyshev expansion

$$arcsinh\left(x\right)=x\times y\left(t\right)=x{\sum ^{\prime }}_{r=0}{c}_r{T}_r\left(t\right)\text{, where}t=2x^2-1\text{.}$$

For $\left|x\right|>1$ it uses the fact that

$$arcsinh\left(x\right)=sign\left(x\right)\times ln(\left|x\right|+\sqrt{x^2+1})\text{.}$$

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

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

$$arcsinh\left(x\right)=sign\left(x\right)\times (ln\left(2\right)+ln\left(\left|x\right|\right))\text{.}$$

References

NIST Digital Library of Mathematical Functions