角向Teukolsky方程在黑洞的引力自作用和准正模等问题的研究中有着重要的应用，目前只能逐级近似求解，影响了人们对黑洞问题的认识。本项目将提出一种获得其精确解的全新方案。运用不同形式的函数变换和变量代换方法，把具有自然边界条件且符合施图姆-刘维尔边值问题的角向Teukolsky方程转化为合流Heun微分方程。根据合流Heun微分方程及其解析解的特性，通过寻找同一本征态的线性相关的两个本征函数构造朗斯基行列式，来确定本征值必须满足的方程。运用Maple软件编程求解朗斯基行列式给出精确的本征值，根据相关参数和本征值进行运算，给出精确的正交归一化的本征函数，绘出在各种量子态下的角分布。最后通过数值计算和作图方法验证它们为角向Teukolsky方程的精确解。本研究成果不仅能够丰富和发展黑洞物理的研究，还可以拓展到电磁场散射理论、原子与分子物理等实际问题中，也可为检验有关实验结果提供理论依据。

The angular Teukolsky equation has played an important role in black holes for the gravitational self-force (GSF), quasi-normal modes (QNM ), radiation, phase transition and others. Unfortunately, the previously approximate solutions have greatly affected our understanding of black holes. The main purpose of this project is to propose an entirely new scheme in order to obtain its exact solutions. First, the angular Teukolsky equation with natural boundary conditions which satisfying the Sturm-Liouville boundary value problem will be transformed to a confluent Heun differential equation by using suitable function transformations and variable substitutions. And then, in terms of the characteristics of the confluent Heun differential equation and its corresponding analytical solutions, two linearly-dependent eigenfunctions which correspond to the same eigenstate are used to construct the Wronskian determinant and then to decide the energy equation of eigenvalues. With the aid of Maple software, we solve the Wronskian determinant and thus obtain its exact eigenvalues. The orthogonal normalized eigenfunctions can be achieved precisely by substituting the related parameters and obtained eigenvalues into non-normalized eigenfunctions. We shall also illustrate the angular distributions for various quantum states. The solutions which are to be verified by numerical calculation and mapping method are shown that they are indeed the exact solutions of the angular Teukolsky equation. The results to be obtained in this project not only enrich the study in black hole physics, but also can be extended to other kinds of practical problems such as electromagnetic field scattering theory, atomic and molecular physics. Moreover, it can also provide theoretical basis for examining relevant experimental results.