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Expansions of the Z function

< PlasmaWiki(Link to this page as [[PlasmaWiki/Expansions of the Z function]])

$Z(\zeta)=\frac{1}{\sqrt{\pi}}\int_{-\infty}^{\infty} \frac{e^{-x^2}dx}{x-\zeta}$

Integration by parts:

$Z'(ζ) = - 2(1 + ζ Z (ζ))$

$\zeta_n = \frac{\omega - k_{\parallel} V-n \Omega}{k_{\parallel} w_{\parallel}}$

Hot: Small $ζ$ , ( $\frac{\omega}{k}<<w_{\parallel}$ )

$Z \simeq i \sqrt{\pi}e^{-\zeta^2}-2\zeta+\frac{4 \zeta^3}{3}-\frac{8 \zeta^5}{15}$

Cold: Large $ζ$ ( $\frac{\omega}{k}>>w_{\parallel}$ )

$Z \simeq i \sqrt{\pi}e^{-\zeta^2}-\frac{1}{\zeta}-\frac{1}{2\zeta^3}-\frac{3}{4 \zeta^5}$