Plasma Dispersion Function

If the unperturbed distribution function, $F_0$, appearing in Equation (7.23), 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

$$Z(\zeta) = \pi^{-1/2}\int_{-\infty}^{\infty} \frac{{\rm e}^{-t^{2}}}{t-\zeta}\,dt,$$ (7.39)

which is defined as it is written for ${\rm Im}(\zeta)>0$, and is analytically continued for ${\rm Im}(\zeta)\leq 0$. This function is known as the plasma dispersion function, and very often crops up in problems involving small-amplitude waves propagating through warm plasmas. Incidentally, $Z(\zeta)$ is the Hilbert transform of a Gaussian function.

In view of the importance of the plasma dispersion function, and its regular appearance in the literature of plasma physics, it is convenient to briefly examine its main properties. We, first of all, note that if we differentiate $Z(\zeta)$ with respect to $\zeta$ then we obtain

$$Z'(\zeta) = \pi^{-1/2}\int_{-\infty}^{\infty} \frac{{\rm e}^{-t^{2}}}{(t-\zeta)^2}\, dt,$$ (7.40)

which yields, on integration by parts,

$$Z'(\zeta) =-\pi^{-1/2}\int_{-\infty}^{\infty} \frac{2\,t}{t-\zeta}\, {\rm e}^{-t^{2}}\,dt = -2\,(1+\zeta\,Z).$$ (7.41)

If we let $\zeta$ tend to zero from the upper half of the complex plane, then we get

$$Z(0) = \pi^{-1/2}\,P\!\int_{-\infty}^{\infty} \frac{{\rm e}^{-t^{2}}}{t}\,dt + {\rm i}\,\pi^{1/2} = {\rm i}\,\pi^{1/2}.$$ (7.42)

Of course, the principal part integral is zero because its integrand is an odd function of $t$.

Integrating the linear differential equation (7.41), which possesses an integrating factor $\exp(\zeta^{2})$, and using the boundary condition (7.42), we obtain an alternative expression for the plasma dispersion function:

$$Z(\zeta) = {\rm e}^{-\zeta^{2}}\left({\rm i}\,\pi^{1/2} -2\!\int_0^\zeta {\rm e}^{x^2}\,dx\right).$$ (7.43)

Making the substitution $t={\rm i}\,x$ in the integral, and noting that

$$\int_{-\infty}^0 {\rm e}^{-t^{2}}\,dt = \frac{\pi^{1/2}}{2},$$ (7.44)

we finally arrive at the expression

$$Z(\zeta) = 2\,{\rm i}\, {\rm e}^{-\zeta^{2}}\int_{-\infty}^{\,{\r... ...rm i}\,\pi^{1/2}\,{\rm e}^{-\zeta^{2}}\left[1+{\rm erf}({\rm i}\,\zeta)\right].$$ (7.45)

This formula, which relates the plasma dispersion function to an error function of imaginary argument (Abramowitz and Stegun 1965), is valid for all values of $\zeta$.

For small $\zeta$, we have the expansion (Richardson 2019)

$$Z(\zeta) = {\rm i}\,\pi^{1/2}\,{\rm e}^{-\zeta^{2}}-2\,\zeta\left... ...\frac{4\,\zeta^{4}}{15} - \frac{8\,\zeta^{6}}{105} +{\cal O}(\zeta^{8})\right].$$ (7.46)

For large $\zeta$, where $\zeta=x+{\rm i}\,y$, the asymptotic expansion for $x>0$ is written (Richardson 2019)

$$Z(\zeta) = {\rm i}\,\pi^{1/2}\,\sigma\,{\rm e}^{-\zeta^{2}} -\zet... ... + \frac{3}{4\,\zeta^{4}}+\frac{15}{8\,\zeta^{6}} +{\cal O}(\zeta^{-8})\right].$$ (7.47)

Here,

$$\sigma = \left\{ \begin{array}{lll} 0&\mbox{\hspace{0.5cm}}& ... ...vert x\vert\\ [0.5ex] 2&&y< -1/\vert x\vert \end{array}\right..$$ (7.48)

In deriving our previous expression (7.32) for the Landau damping rate, we, in effect, used the first few terms of the asymptotic expansion (7.47).

The properties of the plasma dispersion function are specified in exhaustive detail in a well-known book by Fried and Conte (Fried and Conte 1961).