plasma_dispersion_func

plasmapy.mathematics.plasma_dispersion_func(zeta: Union[complex, int, float, numpy.ndarray]) → Union[complex, float, numpy.ndarray]

Calculate the plasma dispersion function.

Parameters:

zeta (complex, int, float, ndarray, or Quantity) – Argument of plasma dispersion function.

Returns:

Z – Value of plasma dispersion function.

Return type:

complex, float, or ndarray

Raises:

• TypeError – If the argument is of an invalid type.
• UnitsError – If the argument is a Quantity but is not dimensionless.
• ValueError – If the argument is not entirely finite.

See also

plasma_dispersion_func_deriv()

Notes

The plasma dispersion function is defined as:

\[Z(\zeta) = \pi^{-0.5} \int_{-\infty}^{+\infty} \frac{e^{-x^2}}{x-\zeta} dx\]

where the argument is a complex number [fried.conte-1961].

In plasma wave theory, the plasma dispersion function appears frequently when the background medium has a Maxwellian distribution function. The argument of this function then refers to the ratio of a wave’s phase velocity to a thermal velocity.

References

[fried.conte-1961]

Fried, Burton D. and Samuel D. Conte. 1961. The Plasma Dispersion Function: The Hilbert Transformation of the Gaussian. Academic Press (New York and London).

Examples

>>> plasma_dispersion_func(0)
1.7724538509055159j
>>> plasma_dispersion_func(1j)
0.757872156141312j
>>> plasma_dispersion_func(-1.52+0.47j)
(0.6088888957234254+0.33494583882874024j)