Fermi_integral

plasmapy.formulary.mathematics.Fermi_integral(x: complex | ndarray, j: complex | ndarray) → complex | ndarray

Calculate the complete Fermi-Dirac integral.

Parameters:

• x (float, int, complex, or ndarray) – Argument of the Fermi-Dirac integral function.

• j (float, int, complex, or ndarray) – Order/index of the Fermi-Dirac integral function.

Returns:

integral – Complete Fermi-Dirac integral for given argument and order.

Return type:

float, complex, or ndarray

Raises:

• TypeError – If the argument is invalid.

• UnitsError – If the argument is a Quantity but is not dimensionless.

• ValueError – If the argument is not entirely finite.

Notes

The complete Fermi-Dirac integral is defined as:

\[F_j (x) = \frac{1}{Γ(j+1)} \int_0^∞ \frac{t^j}{\exp{(t-x)} + 1} dt\]

for \(j > 0\).

This is equivalent to the following polylogarithm function:

\[F_j (x) = -Li_{j+1}\left(-e^{x}\right)\]

Warning

At present this function is limited to relatively small arguments due to limitations in mpmath.polylog.

Examples

>>> Fermi_integral(0, 0)
(0.6931471805599453-0j)
>>> Fermi_integral(1, 0)
(1.3132616875182228-0j)
>>> Fermi_integral(1, 1)
(1.8062860704447743-0j)