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Fast calculation of plasma dispersion functions for Maxwellian distribution and arbitrary distributions.

For arbitrary distributions

A fast and accurate approach is provided, where FFT is used.

Ref and cite: [Xie2013] Hua-Sheng Xie, Generalized plasma dispersion function: One-solve-all treatment, visualizations, and application to Landau damping, Phys. Plasmas 20, 092125 (2013), doi: 10.1063/1.4822332.

Maxwellian distribution (i.e., the most widely used)

The plasma dispersion function $Z(z)$ can also be calculated by J-pole Pade approximation, i.e.,

$Z(z)={\sum}_{j=1}^{J} \frac{b_j}{z-c_j}$.

For J=8, one can use

b1= -0.017340112270401 - 0.046306439626294i;

b2= -0.739917811220052 + 0.839518284620274i;

b3= 5.840632105105495 + 0.953602751322040i;

b4= -5.583374181615043 -11.208550459628098i;

c1= 2.237687725134293 - 1.625941024120362i;

c2= 1.465234091939142 - 1.789620299603315i;

c3= 0.839253966367922 - 1.891995211531426i;

c4= 0.273936218055381 - 1.941787037576095i;

b(5:8)=(b(1:4))*;

c(5:8)=-(c(1:4))*,

where * denotes complex conjugate. The above approximation is valid for the upper plane. For $Im(z)$ weakly close to the real axis, i.e., not far from the real axis of the lower plane, the above approximation is also valid. For accurate calculation of $Im(z)<0$, one can use

$Z(z)=[Z(z*)]*+2i\sqrt{\pi}\exp(-z^2)$.

Other J, such as J=4,8,12,16,24 can also be found. The J-pole method is probably the quickest method to calculate plasma dispersion function, which is accurate better to 1e-6 for J=8, and 1e-13 for J=24. The coding to Fortran and Python is also straightforward.

Ref and cite:

[Xie2019] Huasheng Xie, BO: A unified tool for plasma waves and instabilities analysis, Computer Physics Communications 244 (2019) 343–371.

[Xie2024] H. S. Xie, Rapid Computation of the Plasma Dispersion Function: Rational and Multi-pole Approximation, and Improved Accuracy, AIP Advances 14, 075007 (2024).