I am trying to solve an equation involving a complex argument for the plasma dispersion function as: $z = x + \iota y$,
$ x = \omega / \sqrt2 k v_{Ti} $
$ y = \nu_i /\sqrt{2} k v_{Ti} $
$S[z] = -\iota \sqrt{\pi} Exp[-z^2] Erfc[-\iota z]$;
I have reduced the equation to a system of equations corresponding to real part and imaginary part as:
$(x - \beta)(c_1^2 + c_2^2) + \nu_e(d_2c_2 - d_1c_1)=0$,
$\gamma(c_1^2 + c_2^2) + \nu_e(d_2c_1 - d_1c_2)=0$
I am not able to find how should i solve this system for omega and gamma to obtain plots for $\omega$ v/s k and $\gamma$ v/s k? $c_1, c_2, d_1, d_2$ are all functions of S(z). $\beta, \nu_e, \nu_i, v_{Ti}$ are the parameters whose values are known.