The visualization of the angular probability distribution for the angular Teukolsky equation with m ≠ 0

We present the exact solutions of the angular Teukolsky equation with m. 0 given by a confluent Heun function. This equation is first transformed to a confluent Heun differential equation through some variable transformations. The Wronskian determinant, which is constructed by two linearly dependent solutions, is used to calculate the eigenvalues precisely. The normalized eigenfunctions can be obtained by substituting the calculated eigenvalues into the unnormalized eigenfunctions. The relations among the linearly dependent eigenfunctions are also discussed. When c(2) = c(2) R + i c(2)I, the eigenvalues are approximately expressed as Alm approximate to lol + (l +1)+ (c(R)(2) + ic(I)(2) ) [ 1- m(2)= (/l +1) ] /2 for small jcj2 but large l. The isosurface and contour visualizations of the angular probability distribution (APD) are presented for the cases of the real and complex values c2. It is found that the APD has obvious directionality, but the northern and southern hemispheres are always symmetrical regardless of the value of the parameter c2, which is real or imaginary.