naginterfaces.library.specfun.integral_​cos

naginterfaces.library.specfun.integral_cos(x)

integral_cos returns the value of the cosine integral

$$Ci\left(x\right)=\gamma +ln\left(x\right)+{\int }_0^x\frac{\cos \left(u\right)-1}{u}du\text{,}x>0$$

where $\gamma$ denotes Euler’s constant.

For full information please refer to the NAG Library document for s13ac

https://support.nag.com/numeric/nl/nagdoc_30.3/flhtml/s/s13acf.html

Parameters

xfloat

The argument $x$ of the function.

Returns

cixfloat

The value of the cosine integral

$$Ci\left(x\right)=\gamma +ln\left(x\right)+{\int }_0^x\frac{\cos \left(u\right)-1}{u}du\text{,}x>0\text{.}$$

Raises

NagValueError

(errno $1$)

On entry, $x=⟨value⟩$.

Constraint: $x>0.0$.

The function has been called with an argument less than or equal to zero for which $Ci\left(x\right)$ is not defined.

Notes

integral_cos calculates an approximate value for $Ci\left(x\right)$.

For $0<x\le 16$ it is based on the Chebyshev expansion

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

For $16<x<x_{hi}$,

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

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 $x\ge x_{hi}$, $Ci\left(x\right)=0$ to within the accuracy possible (see Accuracy).

References

NIST Digital Library of Mathematical Functions