naginterfaces.library.specfun.arctanh

naginterfaces.library.specfun.arctanh(x)

arctanh returns the value of the inverse hyperbolic tangent, $arctanh\left(x\right)$.

For full information please refer to the NAG Library document for s11aa

https://support.nag.com/numeric/nl/nagdoc_30.3/flhtml/s/s11aaf.html

Parameters

xfloat

The argument $x$ of the function.

Returns

athxfloat

The value of the inverse hyperbolic tangent, $arctanh\left(x\right)$.

Raises

NagValueError

(errno $1$)

On entry, $x=⟨value⟩$.

Constraint: $\left|x\right|<1$.

The function has been called with an argument greater than or equal to $1.0$ in magnitude, for which $arctanh$ is not defined.

Notes

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

For $x^2\le \frac{1}{2}$ it is based on the Chebyshev expansion

$$arctanh\left(x\right)=x\times y\left(t\right)=x{\sum ^{\prime }}_{r=0}{a}_r{T}_r\left(t\right)$$

where $-\frac{1}{\sqrt{2}}\le x\le \frac{1}{\sqrt{2}}$, $-1\le t\le 1$, $\text{and}t=4x^2-1$.

For $\frac{1}{2}<x^2<1$, it uses

$$arctanh\left(x\right)=\frac{1}{2}ln\left(\frac{1+x}{1-x}\right)\text{.}$$

For $\left|x\right|\ge 1$, the function fails as $arctanh\left(x\right)$ is undefined.

References

NIST Digital Library of Mathematical Functions