# plasma_dispersion_func¶

plasmapy.dispersion.dispersionfunction.plasma_dispersion_func(zeta: Union[complex, int, float, numpy.ndarray, astropy.units.quantity.Quantity]) → Union[complex, float, numpy.ndarray, astropy.units.quantity.Quantity]

Calculate the plasma dispersion function.

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

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 [1].

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

 [1] Fried, Burton D. and Samuel D. Conte. 1961. The Plasma Dispersion Function: The Hilbert Transformation of the Gaussian. Academic Press (New York and London). ISBN 9781483261737

Examples

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