[en] This work develops an extremely general electrostatic dispersion relation for a collisionless plasma and codes it for a computer. Once this was completed, a computer code for the plasma dispersion function was generated, since it too formed an integral part of the work. Since the completed code was bulky, it was used to generate previously published results several times to ensure confidence in its accuracy. When these were successfully demonstrated, several current problems in plasma physics were analyzed. A mechanism was suggested for the excitation of electrostatic ion-cyclotron waves at frequencies below the ion-cyclotron frequency in the midst of the auroral acceleration region, which is assumed to consist of downward-moving double layers. The mechanism involves upward flowing ions interacting with a downward flowing background plasma. Another situation studied was waves at frequencies below the lower hybrid frequency being triggered by a highly energetic low density electron beam in a multi-species plasma. The excited waves are either ion-cyclotron-resonance cone waves or beam-plasma waves. The last case studied was counterstreaming bulk ion drift against neutralizing stationary background electrons