naginterfaces.library.specfun.sinh

naginterfaces.library.specfun.sinh(x)

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

For full information please refer to the NAG Library document for s10ab

https://support.nag.com/numeric/nl/nagdoc_30.3/flhtml/s/s10abf.html

Parameters

xfloat

The argument $x$ of the function.

Returns

shxfloat

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

Warns

NagAlgorithmicWarning

(errno $1$)

On entry, $x=⟨value⟩$.

Constraint: $\left|x\right|\le E_1$.

The function has been called with an argument too large in absolute magnitude. There is a danger of overflow. The result returned is the value of $\sinh \left(x\right)$ at the closest argument for which a valid call could be made.

Notes

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

For $\left|x\right|\le 1$ it uses the Chebyshev expansion

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

where $t=2x^2-1$.

For $1<\left|x\right|\le E_1\text{,}\sinh \left(x\right)=\frac{1}{2}(e^x-e^{-x})$

where $E_1$ is a machine-dependent constant.

For $\left|x\right|>E_1$, the function fails owing to the danger of setting overflow in calculating $e^x$. The result returned for such calls is $\sinh \left(sign\left(x\right)E_1\right)$, i.e., it returns the result for the nearest valid argument.

References

NIST Digital Library of Mathematical Functions