naginterfaces.library.specfun.integral_​sin

naginterfaces.library.specfun.integral_sin(x)

integral_sin returns the value of the sine integral

$$Si\left(x\right)={\int }_0^x\frac{\sin \left(u\right)}{u}du\text{,}$$

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For full information please refer to the NAG Library document for s13ad

https://support.nag.com/numeric/nl/nagdoc_30.3/flhtml/s/s13adf.html

Parameters

xfloat

The argument $x$ of the function.

Returns

sixfloat

The value of the sine integral

$$Si\left(x\right)={\int }_0^x\frac{\sin \left(u\right)}{u}du\text{.}$$

Notes

integral_sin calculates an approximate value for $Si\left(x\right)$.

For $\left|x\right|\le 16.0$ it is based on the Chebyshev expansion

$$Si\left(x\right)=x\sum _{r=0}^{\prime }{a}_r{T}_r\left(t\right),t=2{\left(\frac{x}{16}\right)}^2-1\text{.}$$

For $16<\left|x\right|<x_{hi}$, where $x_{hi}$ is an implementation-dependent number,

$$Si\left(x\right)=sign\left(x\right)\{\frac{\pi }{2}-\frac{f\left(x\right)\cos \left(x\right)}{x}-\frac{g\left(x\right)\sin \left(x\right)}{x^2}\}$$

where $f\left(x\right)={{\sum }^{\prime }}_{r=0}{f}_r{T}_r\left(t\right)$ and $g\left(x\right)={{\sum }^{\prime }}_{r=0}{g}_r{T}_r\left(t\right)$, $t=2{\left(\frac{16}{x}\right)}^2-1$.

For $\left|x\right|\ge x_{hi}$, $Si\left(x\right)=\frac{1}{2}\pi sign\left(x\right)$ to within machine precision.

References

NIST Digital Library of Mathematical Functions