naginterfaces.library.specfun.integral_​exp

naginterfaces.library.specfun.integral_exp(x)

integral_exp returns the value of the exponential integral $E_1\left(x\right)$.

For full information please refer to the NAG Library document for s13aa

https://support.nag.com/numeric/nl/nagdoc_30.2/flhtml/s/s13aaf.html

Parameters

xfloat

The argument $x$ of the function.

Returns

e1xfloat

The value of the exponential integral $E_1\left(x\right)$.

Raises

NagValueError

(errno $2$)

On entry, $x=⟨value⟩$ and the constant $x_{hi}=⟨value⟩$. The evaluation has been abandoned due to the likelihood of overflow.

Constraint: $x\ge -x_{hi}$.

Warns

NagAlgorithmicWarning

(errno $1$)

On entry, $x=0.0$ and the function is infinite.

Notes

integral_exp calculates an approximate value for

$$E_1\left(x\right)=-Ei(-x)={\int }_x^{\infty }\frac{e^{-u}}{u}du\text{.}$$

using Chebyshev expansions, where $x$ is real. For $x<0$, the real part of the principal value of the integral is taken. The value $E_1\left(0\right)$ is infinite, and so, when $x=0$, integral_exp exits with an error and returns the largest representable machine number.

For $0<x\le 4$,

$$E_1\left(x\right)=y\left(t\right)-ln\left(x\right)={\sum ^{\prime }}_r{a}_r{T}_r\left(t\right)-ln\left(x\right)\text{,}$$

where $t=\frac{1}{2}x-1$.

For $x>4$,

$$E_1\left(x\right)=\frac{e^{-x}}{x}y\left(t\right)=\frac{e^{-x}}{x}{\sum ^{\prime }}_r{a}_r{T}_r\left(t\right)\text{,}$$

where $t=-1.0+\frac{14.5}{(x+3.25)}=\frac{11.25-x}{3.25+x}$.

In both cases, $-1\le t\le +1$.

For $x<0$, the approximation is based on expansions proposed by Cody and Thatcher Jr. (1969). Precautions are taken to maintain good relative accuracy in the vicinity of $x_0\approx -0.372507\dots$, which corresponds to a simple zero of Ei($-x$).

integral_exp guards against producing underflows and overflows by using the argument $x_{hi}$. To guard against overflow, if $x<-x_{hi}$ the function terminates and returns the negative of the largest representable machine number. To guard against underflow, if $x>x_{hi}$ the result is set directly to zero.

References

NIST Digital Library of Mathematical Functions

Cody, W J and Thatcher Jr., H C, 1969, Rational Chebyshev approximations for the exponential integral Ei $\left(x\right)$, Math. Comp. (23), 289–303