### 6.1 Scalar wave equation in Kerr–Newman

Let us first discuss probe scalar fields on the Kerr–Newman black hole. The Klein–Gordon equation for a
charged massive spin 0 field of mass and charge was analyzed in Section 5.2. Expanding in
eigenmodes and using the fact that the equation is separable, we have
The equations for the functions and were written in (158) and (159). Substituting
, the equations can be written in a convenient way as
where . The function is defined as
and is evaluated either at or and
These equations can be solved by Heun functions, which are not among the usual special functions. A
solution can be found only numerically.