应用伪解析函数论的新进展
Recent developments in applied pseudoanalytic function theory
摘要：
本文介绍了伪解析函数理论及其在二阶椭圆型方程中的应用。利用特殊形式的Vekua型算子对p和q为实值函数的算子（divpgrad+q）进行分解，从而将椭圆型方程（divpgrad+q）u=0，（1）化为描述广义解析（或伪解析）函数的齐次Vekua方程.作为求解Vekua方程的工具，我们利用L. Bers该级数具有通常解析幂级数的许多重要性质。它们的应用直到最近才受到限制，主要是因为它们不可能在一般情况下明确解释。我们得到了一个算法，在一个真正广泛的实际应用范围内，使我们能够构造形式的权力，从而伪解析泰勒级数的显式形式，精确地为Vekua方程有关的方程（1）。换句话说，在一个有界区域中，这给出了（1）的精确解的完备（C-范数）系统。
英文摘要：
We present recently obtained results in the theory of pseudoanalytic functions and its applications to elliptic second-order equations. The operator (divpgrad+q) with p and q being real valued functions is factorized with the aid of Vekua type operators of a special form and as a consequence the elliptic equation (divpgrad+q)u=0, (1) reduces to a homogeneous Vekua equation describing generalized analytic (or pseudoanalytic) functions. As a tool for solving the Vekua equation we use the theory of Taylor and Laurent series in formal powers for pseudoanalytic functions developed by L. Bers. The series possess many important properties of the usual analytic power series. Their applications until recently were limited mainly because of the impossibility of their explicit construction in a general situation. We obtain an algorithm which in a really broad range of practical applications allows us to construct the formal powers and hence the pseudoanalytic Taylor series in explicit form precisely for the Vekua equation related to equation (1). In other words, in a bounded domain this gives us a complete (in C-norm) system of exact solutions of (1).
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
