4.6 Property of

In this section, we discuss the property of the solution of Equation (136) which we recapitulate:
The numerical evaluation of this equation was not done very much before. Leaver [64] briefly mentioned a numerical implementation of a code to obtain by solving Equation (174). In the Schwarzschild case, Tagoshi and Nakamura [99] solved Equation (174) numerically to obtain . They evaluated the homogeneous solution numerically based on Leaver’s method [64] by using the value of obtained numerically. Later, Fujita and Tagoshi [40] tried a numerical implementation of the MST method. They found that, as becomes large, it becomes impossible to obtain a solution of (174) if is restricted to a real number. In a subsequent paper [41], they found that when the real solution ceases to exist, a complex solution appears. They also found that when is complex, the real part of is always either an integer or half-integer. As an example, we show the value of as a function of in Table 1.

The fact that we only have an integer or half-integer as the real part of is strongly suggested from the property of  [37]. Let us summarize the argument. We first convert to as , where is an arbitrary integer and is an arbitrary complex number. We note that for an arbitrary integer ,

where denotes complex conjugation. From these relations, we find that if is a solution of , we have
where we used the relation, . We see that in this case is also an solution of . As already discussed at the end of Section 4.2, when is a solution of , and with an arbitrary integer are also solutions. We assume that there are no other solutions. Although we do not have a formal proof of it, numerical investigations suggest that this is so. Under this assumption, since both and are solutions of , we have two possibilities about the property of :
In the former case, is real. In this case, is complex with real part (integer or half-integer). In the later case, is pure imaginary. In this case, is real.

It becomes possible to determine in the wide range of by allowing . The MST formalism is now very useful in the fully numerical evaluation of homogeneous solutions of the Teukolsky equation. As a first step, Fujita, Hikida and Tagoshi [38] considered generic bound geodesic orbits around a Kerr black hole and evaluate the energy and angular momentum flux to infinity as well as the rate of change of the Carter constant in a wide range of orbital parameters.

The critical value of when becomes complex is not very small. The complex does not appear in the analytic evaluation of in the low frequency expansion in powers of . Thus, at the first glance, it seems impossible to express the complex in the power series expansion of . However, Hikida et al. [52] pointed out that it is possible to evaluate very accurately in terms of the power series expansion of , even if is larger than a critical value and is complex. Such an analytical expression of is very useful in the numerical root finding of Equation (174) as well as in the analytical calculation of the homogeneous solutions.

Table 1: The value of for various value of in the case , and .
 Re Im 0.1 1.9793154547208 0.0000000000000 0.2 1.9129832302687 0.0000000000000 0.3 1.7792805424199 0.0000000000000 0.4 1.5000000000000 0.1862468531447 0.5 1.5000000000000 0.3618806153941 0.6 1.7878302655744 0.0000000000000 0.7 2.0000000000000 0.8003377636925 0.8 2.0000000000000 1.1099466644118 0.9 2.0000000000000 1.3699138540831 1.0 2.0000000000000 1.6085538776570 2.0 2.0000000000000 3.6867890278893 3.0 2.0000000000000 5.5939000509184