Classes of Exact Solutions to the Teukolsky Master Equation

The Teukolsky Master Equation describes perturbations of the Kerr metric in linear approximation. It admits separation of variables, thus yielding the Teukolsky Radial Equation and the Teukolsky Angular Equation. We present here a unified description of all classes of exact solutions to these equations in terms of the confluent Heun functions and the confluent Heun polynomials. Large classes of new exact solutions are found and described together with their characteristic properties. Special attention is paid to the polynomial solutions which are singular ones and describe collimated one-wayrunning waves. It is shown that a proper linear combination of such singular solutions can describe bounded one-way-running waves.