Adv. Appl. Cliﬀord Algebras 27 (2017), 2531–2583
2017 Springer International Publishing
published online May 29, 2017
Applied Cliﬀord Algebras
Generalized Cauchy Theorem in Cliﬀord
Analysis and Boundary Value Problems
for Regular Functions
Weiyu Luo and Jinyuan Du
Communicated by Frank Sommen
Abstract. In this paper, we establish the generalized Cauchy theorems
on the para-sphere and the generalized Cauchy integral formulae on the
strong para-sphere in Cliﬀord analysis. As applications, the generalized
Cauchy theorems and the generalized Cauchy integral formulae on the
closed smooth surface and the cylindroid with crooked tips are respec-
tively obtained. And these directly result in the Painlev´e theorem and
the generalization of the Sochocki–Plemelj formula for the diﬀerence of
boundary values in Cliﬀord analysis. Then, by using these results the
Riemann jump boundary value problems and Dirichlet boundary value
problems for regular functions in Cliﬀord analysis are discussed. Some
singular integral equations are also solved and the inversion formula
for Cauchy principal value is obtained by the results based on these
boundary value problems solved.
Mathematics Subject Classiﬁcation. 30G35, 30E20, 30E25.
Keywords. Generalized Cauchy theorem, Generalized Cauchy integral
formula, Painlev´e theorem, Riemann boundary value problem, Singular
integral equation, Dirichlet boundary value problem.
As we all know, Cauchy’s Theorem and Cauchy’s Integral Formula are the
basic results of the classical complex analysis [1,13]. Their applications cover
all branches in this ﬁeld, even their generalized versions have to be used in
some branches [35,39], such as, boundary value problems (BVPs) for ana-
lytic functions [21,32,37]. The BVPs for analytic functions is an important
This work was supported by NNSF of China (#11171260) and RFDP of Higher Education
of China (#20100141110054).