Analytic dispersion solver of cold plasma waves with computer algebra SymPy

We present computer algebra of the polynomial dispersion equation for the cold plasma waves (neglecting the displacement current) using the Python library SymPy. We obtain the dispersion equation as a polynomial of the frequency by compensating for singularities in the dielectric tensor, such as the zero frequency and the cyclotron resonance, and further by eliminating the unphysical modes. The polynomial dispersion equation has a wavenumber degree of four and a frequency degree of two times the number of species, e.g., fourth-order polynomial for the electron–proton plasma, corresponding to the whistler and ion-cyclotron modes in the positive and negative frequency domains. The polynomial equation gives analytic forms of the dispersion relation for the whistler and ion-cyclotron modes for arbitrary propagation directions to the mean magnetic field, which serves as a useful tool in various research fields involving plasma waves. We provide the SymPy dispersion solver and the numerical dispersion solver (Stix method) as open-access software.