darkhistory.electrons.ics.BE_integrals.F1

darkhistory.electrons.ics.BE_integrals.F1(a, b, epsrel=0)

Definite integral of x/[(exp(x) - 1)].

This is computed from the indefinite integral

\[\int dx \frac{x}{e^x - 1} = x \log\left(1 - e^{-x} \right) - \text{Li}_2\left(e^{-x}\right) = x \log\left(1 - e^{-x} \right) - \text{Sp}\left( 1 - e^{-x} \right) + \frac{\pi^2}{6} \,,\]

where Sp is Spence’s function, as implemented in scipy.special.spence.

Parameters:

a : ndarray

Lower limit of integration. Can be either 1D or 2D.

b : ndarray

Upper limit of integration. Can be either 1D or 2D.

epsrel : float

Target relative error associated with series expansion. If zero, then the error is not computed. Default is 0. If the error is larger than epsrel, then the Taylor expansions used here are insufficient. Higher order terms can be added very easily, however.

Returns:

float

The resulting integral.

See also

log_1_plus_x(), spence_series_diff()

Notes

For a or b > 0.01, the exact analytic expression is used, whereas below that we use a series expansion. This avoids numerical errors due to computation of log(1 - exp(-x)) and likewise in the spence function. Note that scipy.special.spence can only take float64 numbers, so downcasting is necessary for 0.01 < x < 3.