四元数分析中超球与双圆柱区域上的正则函数

本文讨论了四元数分析中的正则函数Ｕ（ｚ）（满足方程ｚＵ（ｚ）＝０，ｚ＝ｘ１＋ｉｘ２＋ｊｘ３－ｋｘ４）及其边值问题，给出了超球与双圆柱区域上的四元数正则函数的Ｃａｕｃｈｙ积分公式，获得了一般区域上正则函数的无穷次可微性；给出了定义在超球与双圆柱区域边界上的四元数函数可正则开拓到区域内的条件；讨论了满足非齐次方程ｚＦ＝ｆ的四元函数Ｆ（ｚ）的Ｄｉｒｉｃｈｌｅｔ和Ｎｅｕｍａｎｎ边值问题；获得了超球与双圆柱区域上这两种边值问题解的积分表示．

REGULAR FUNCTION ON THE BALL AND THE BICYLINDER IN QUATERNION CALCULUS

in this paper, we obtain the Cauchy integral formulas of the regular quaternion functions on the ball and bicylinder, and prove the infinite differentiability of the regularquaternion fUnction on the general domain. Conditions for a quaternion function defined onthe boundary of the ball or the bicylinder to be able to be extended regularly into the inside ofthe domain are derived. We also discuss the Dirichlet and Neumann boundary value problemsfor the quaternion function F(z) satisfying the nonhomogeneous eqaution zF = f, and forthese problems have obtained the integral expressions of solutions on the ball and bicylinder.