Plasma dispersion function

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The plasma dispersion function is defined for Im(ζ) > 0 as:

Z(\zeta)=\frac{1}{\sqrt{\pi}}\int_{-\infty}^{\infty}dt\frac{e^{-t^2}}{t-\zeta}

Where:

\zeta=\frac{\omega}{k v_T}

with vT the thermal velocity .

For |\zeta|\ll 1:

Z(\zeta)=i\pi^{1/2} e^{-\zeta^2} - 2\zeta \left(1-\frac{2 \zeta^2}{3} + \frac{4 \zeta^4}{15} - \frac{8 \zeta^6}{105} + \cdots \right)

For |\zeta|\gg 1:

Z(\zeta)=i\pi^{1/2} \sigma e^{-\zeta^2} - \zeta^{-1} \left(1+\frac{1}{2 \zeta^2} + \frac{3}{4 \zeta^4} + \frac{15}{8 \zeta^6} + \cdots \right)

With ζ = x + iy:

Failed to parse (PNG conversion failed; check for correct installation of latex, dvips, gs, and convert): \sigma = \begin{cases} 0, &amp; \mbox{if } y &gt; |x|^{-1} \\ 1, &amp; \mbox{if } |y| &lt; |x|^{-1} \\ 2, &amp; \mbox{if } y &lt; -|x|^{-1} \end{cases}

[1] B.D. Fried & S.A. Conte, "Plasma Dispersion Function", (Book, 1961, Academic Press, NY).

This page was recovered in October 2009 from the Plasmagicians page on Plasma_dispersion_function dated 23:43, 14 May 2007.

Retrieved from "https://qed.princeton.edu/main/PlasmaWiki/Plasma_dispersion_function"
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