A Generalization of Hartogs Theorem

Abstract

Publisher Summary This chapter discusses the generalization of Hartogs theorem. In the theory of complex functions of several complex variables, there is a basic result of Hartogs that a separately analytic function is analytic. From the point of view of partial differential equations, this may be stated as a separately continuous differentiable function u of two independent complex variables, z 1 and z 2 , which satisfies the Cauchy-Riemann equations is analytic in both variables z 1 and z 2 together. This chapter discusses the variants of a famous theorem on sequences of analytic functions of one complex variable, due to Hartogs. A specific form of Schwarz’ lemma is discussed in the chapter.