广义多解析函数的多种积分表示及应用

Integral Representations for Generalized Poly-Analytic Functions and its Application

积分表示作为复分析基本理论和研究边值问题的重要工具,基于此提出并建立一类更广泛的广义多解析函数类的积分表示及应用理论。利用广义β-解析函数的分解定理,并结合Cauchy-Pompeiu公式、矩阵变换技巧和Fredholm积分方程理论进行研究。获得了包含带平移和不带平移在内的多种广义多解析函数的积分表示式,并由此定义并证明了高阶多Cauchy型积分的连续延拓定理及该定理在求解一类黎曼跳跃问题的应用。建立了一类广义多解析函数的积分表示,延伸和推广了解析函数特别是多解析函数的积分表示理论,也为后续相关多β-解析函数边值问题和奇异积分算子等研究提供了理论支撑。

On account of integral representations regarded as the basic theory of complex analysis and a significant tool for solving boundary value problems,integral representations for a class of broader generalized poly-analytic functions are proposed and established.By using decomposition theorem and Cauchy-Pompeiu formula of generalizedβ-analytic functions,together with matrix transformation and the theory of Fredholm integral equations,various integral representations are investigated.Various integral representations including with shift and without shift are obtained,the extension theorem of higher order poly-Cauchy type integral is proved,and the application in solving a class of Riemann jump problems is also provided.Several types of integral representations for a class of generalized poly-analytic function are established,which extend and generalize the integral representation theory of analytic functions,especially poly-analytic functions,and also provide theoretical support for the research on boundary value problems and singular integral operators related to β-analytic functions in the future.