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If the unperturbed distribution function, , appearing in Eq. (1007), is
a Maxwellian then it is readily seen that, with a suitable scaling of the
variables, the dispersion relation for electrostatic
plasma waves can be expressed in terms of the
function

(1022) 
which is defined as it is written for
, and is
analytically continued for
. This function is
known as the plasma dispersion function, and very often crops up
in problems involving smallamplitude waves propagating through
warm plasmas. Incidentally, is the Hilbert transform of a Gaussian.
In view of the importance of the plasma dispersion function, and its regular
occurrence in the literature of plasma physics, let us briefly examine its main
properties. We first of all note that if we differentiate with
respect to we obtain

(1023) 
which yields, on integration by parts,

(1024) 
If we let tend to zero from the upper half of the complex plane, we get

(1025) 
Note that the principle part integral is zero because its integrand is an
odd function of .
Integrating the linear differential equation (1024), which possesses an
integrating factor
, and using the boundary condition
(1025), we obtain an alternative expression for the plasma dispersion
function:

(1026) 
Making the substitution in the integral, and
noting that

(1027) 
we finally arrive at the expression

(1028) 
This formula, which relates the plasma dispersion function to an
error function of imaginary argument, is valid for all values
of .
For small we have the expansion

(1029) 
For large , where
, the asymptotic expansion
for is written

(1030) 
Here,

(1031) 
In deriving our expression for the Landau damping rate we have, in effect, used the
first few terms of the above asymptotic expansion.
The properties of the plasma dispersion function are specified in exhaustive
detail in a wellknown book by Fried and Conte.^{}
Next: Ion Sound Waves
Up: Waves in Warm Plasmas
Previous: Physics of Landau Damping
Richard Fitzpatrick
20110331