Fueter定理在八元数中的推广

The Generalization of Fueter Theorem on Octonions

设f(z)=u(s,t)+iv(s,t)为上半平面某区域上的复解析函数,将s,t与i分别替换为x0,r=|x|=√x1^2+x2^2+…+x7^2与ω=x/|x|=x1e1+x2e2+…+x7e7/r,得到八元数值诱导函数F(x)=u(x0,r)+ωv(x0,r),给出Δ8^k+3[F(x)Pk(x)]为O-解析函数的详细证明,其中Pk(x)为k阶齐次O-解析函数.这是Fueter定理在八元数中的推广.

Let f(z)=u(s,t)+iv(s,t) be a complex analytic functions in some domain of the upper half plane,replace s,t and i with x 0,r=|x|=√x1^2+x2^2+…+x7^2 and ω=x/|x|=x1e1+x2e2+…+x7e7/r ,then get the octonion induced function F(x)=u(x 0,r)+ω v(x 0,r).This paper gives a detailed proof for Δ8^k+3[F(x)P k(x )] being a left O -analytic function,where P k(x ) is a homogeneous O -analytic function of order k.This is a generalization of Fueter theorem on octinions.