四元数空间中的正则函数与某些边值问题

REGULAR FUNCTION AND SOME BOUNDARY VALUE PROBLEMS IN THE QUATERNION SPACE

本文讨论四元数（有单位元１，ｉ，ｊ，ｋ，ｉ２＝ｊ２＝ｋ２＝－１，ｉｊ＝ｋ＝－ｊｉ）正则函数与正则调和函数的关系，首先证明了数量调和函数的共轭矢量调和函数的存在性及矢量调和函数存在共轭数量调和函数的充要条件；其次证明了广义多圆柱区域上正则函数的Ｄｉｒｉｃｈｌｅｔ边值问题的可解性并给出了通解表达式；最后讨论了一个非齐次方程 Ｕ＝ＡＵ＋Ｂ ＋Ｃ的Ｄｉｒｉｃｈｌｅｔ边值问题的可解性。

This paper discusses the relation between the regular quaternion function and the regular harmonic function and some boundary value problems in the quaternion calculus(with units:1,i,j,k,i2=j2=k2=-1,ij =k=-ji).We prove the existence of conjugate vector harmonic functions of the scalar harmonic function and obtain the necessary and sufficient condition that the conjugate scalar harmonic function of a vector harmonic function exists;prove the solvability of the Dirichlet boundary value problem for regular quaternion functions in a generalized polycylinder and obtain its expressions of solution;prove that the solvability of the Dirichlet boundary value problem for the solutions of quaternion equation depends on the solvabilities of three real second order elliptic partial differential equations.