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Analytic solutions of the Teukolsky equation in Kerr geometries are presented in the form of series of hypergeometric functions and Coulomb wave functions. Relations between these solutions are established. The solutions provide a very powerful method not only for examining the general properties of solutions and physical quantities when they are applied to, but also for numerical computations. The solutions are given in the expansion of a small parameter ǫ ≡ 2Mω, M being the mass of black hole, which corresponds to Post-Minkowski expansion by G and to post-Newtonian expansion when they are applied to the gravitational radiation from a particle in circular orbit around a black hole. It is expected that these solutions will become a powerful weapon to construct the theoretical template towards LIGO and VIRGO projects.

It is well known that the perturbation equations of massless fields for the Kerr-de Sitter geometry can be written in the form of separable equations. The equations have five definite singularities, so it has been thought that their analysis would be difficult. We show that these equations can be transformed into Heun's equations, for which we are able to use a known technique to analyze solutions. We reproduce known results for the Kerr geometry and de Sitter geometry in the confluent limits of Heun's functions. Our analysis can be extended to Kerr-Newman-de Sitter geometry for massless fields with spin 0 and ~. §1. IntroductionOne of the most non-trivial aspects of the perturbation equations for Kerr geometry is the separability of the radial and the angular parts. Carter 1) first found that the scalar wave function is separable in the Kerr-Newman-de Sitter geometries. Later, this observation was extended for spin 1/2, electromagnetic fields, gravitational perturbations and gravitinos for the Kerr geometries and even for the Kerr-de Sitter class of geometries. These perturbation equations are called Teukolsky equations. 2 ) Except for electromagnetic and gravitational perturbations, the separability persists even for the Kerr-Newman-de Sitter solutions. An important application of this fact is the proof of the stability of the Kerr black hole. 3) Though the Teukolsky equations for Kerr geometries are separable, both spheroidal and radial equations have two regular singularities and one irregular singularity, so that the solutions cannot be written in a single form of any special functions, but they can be expressed as a series of special functions whose coefficients satisfy three term recurrence relations. 4) -7) The solution of the angular equation is expressed in the form of a series of Jacobi functions. 4 ) The solution of the radial equation is rather complicated, because we need a solution which is valid in the entire region extending from the outer horizon to infinity. The solutions are written in the form of a series of confluent hypergeometric functions 5) which are convergent around infinity and hypergeometric functions 6), 7) which are convergent around the outer horizon. By matching these solutions in the region where both solutions are convergent, we can obtain a solution which is valid from the outer horizon to infinity. 6)-8) The great benefit of this kind of solution is that the coefficients of the series are obtained through the post Minkowskian expansion. 6), 7) This technique has been *)

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549 Analytic solutions of the Regge·Wheeler equation are presented in the form of a series of hypergeometric functions and Coulomb wave functions which have different regions of convergence. Relations among these solutions are established. The series solutions are given as the PostMinkowskian expansion with respect to the parameter €=2Mw, M being the mass of a black hole.This expansion corresponds to the post-Newtonian expansion when they are applied to the gravitational radiation from a particle in a circular orbit around a black hole. These solutions can also be useful for numerical computations. § 1. IntroductionIn a previous work,l) we presented analytic solutions of the Regge-Wheeler (RW) equation in the form of a series of hypergeometric functions. We proved that recurrence relations among hypergeometric functions as given in Appendix A in this text and showed that coefficients of series are systematically determined in a power series of €=2Mw, M being the mass of black hole. We also presented analytic solutions in the form of a series of Coulomb wave functions which turn out to be the same as those given by Leaver. 2 ) We found that the series of solutions is characterized by the renormalized angular momentum which turns out to be identical. Then, we obtained a good solution by matching these two types of solutions.This method can be extended for the Teukolsky equation 3 ) in the Kerr geometry_ In this case, the coefficients of series of hypergeometric functions and also those of a series of Coulomb wave functions satisfy the three term recurrence relations. Concerning these recurrence relations, Otchik 4 ) made the important observation that the recurrence relation for the two series are identical, which made it possible to relate these two series solutions_*) Following the discussion by Otchik,4) Mano, Suzuki and Takasugi 5 ) extended our analysis to the Teukolsky equation in the Kerr geometry and reported analytic solutions. We discussed the convergence regions of these series and the relation between two solutions of different regions of convergence. The series are expressed in the € expansion which corresponds to the Post-Minkowskian expansion and also to the post-Newtonian expansion when they are applied to the gravitational radiation from a particle in circular orbit around a black hole.In this paper, we present analytic solutions of the RW equation and discuss the analytic properties of these solutions by reorganizing our previous work!) following *) In Otchik's paper, the relation between the series of hypergeometric functions and the series of Coulomb wave functions is studied in the intermediate region where both series converge, though the series which he treated are not the solutions of Teukolsky equation.