Modified plasma dispersion function

The modified plasma dispersion function introduced by Summers and Thorne are defined as

\[Z_{\kappa_{\|}}\left(\xi_n\right)=\frac{\Gamma\left(\kappa_{\|_s}+1\right)}{\sqrt{\pi \kappa_{\|_s^3}^3 \Gamma}\left(\kappa_{\|_s}-1 / 2\right)} \int_{-\infty}^{\infty} \frac{d x}{\left(x-\xi_n\right)\left(1+x^2 / \kappa_{\|_s}\right)^{\kappa_{\| s}+1}}, \quad \Im(\xi)>0\]

with $x=\left(v_{\|}-u_{s 0}\right) / \theta_{\| s}$ and $\xi_n=\left(\omega-n \omega_{c s}-k_{\|} u_{s 0}\right) / k_{\|} \theta_{\| s}$. For any positive integer $\kappa_{\|_s}$, and all $\xi_n \neq \pm i \sqrt{\kappa_{\| s}}$, the expressions for these dispersion functions can be given using closed-form expansions,

\[Z_{\kappa_{\|}}\left(\xi_n\right)=\frac{i\left(\kappa_{\| s}-1 / 2\right)}{2 \kappa_{\| s}^{3 / 2}} \frac{\kappa_{\| s}!}{\left(2 \kappa_{\|_s}\right)!} \sum_{l=1}^{\kappa_{\| s}+1} \frac{\left(\kappa_{\| s}+l-1\right)!}{(l-1)!}\left(\frac{2 i}{\left(\xi_n / \sqrt{\kappa_{\|_s}}\right)+i}\right)^{\kappa_{\| s}-l+2} .\]