naginterfaces.library.specfun.integral_​exp

naginterfaces.library.specfun.integral_exp(x)[source]

integral_exp returns the value of the exponential integral .

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

https://www.nag.com/numeric/nl/nagdoc_28.5/flhtml/s/s13aaf.html

Parameters
xfloat

The argument of the function.

Returns
e1xfloat

The value of the exponential integral .

Raises
NagValueError
(errno )

On entry, and the constant . The evaluation has been abandoned due to the likelihood of overflow.

Constraint: .

Warns
NagAlgorithmicWarning
(errno )

On entry, and the function is infinite.

Notes

integral_exp calculates an approximate value for

using Chebyshev expansions, where is real. For , the real part of the principal value of the integral is taken. The value is infinite, and so, when , integral_exp exits with an error and returns the largest representable machine number.

For ,

where .

For ,

where .

In both cases, .

For , 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 , which corresponds to a simple zero of Ei().

integral_exp guards against producing underflows and overflows by using the argument . To guard against overflow, if the function terminates and returns the negative of the largest representable machine number. To guard against underflow, if 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 , Math. Comp. (23), 289–303