Weierstrass Complex Function Of Ideological Analysis

By means of methods of text-analysis, historical and comparative research, the dissertation carries on detailed, comprehensive and systematical interpretation and analysis based on intensive investigation of original treatises and lectures of Weierstrass and other related research literature. With the purport of exploring fundamental questions- the shaping and development of Weierstrass' thought, method and theory in complex function, and combining with the theory of function and real analysis, theoretical construction of Weierstrass' complex function are analyzed and regenerated, and the mathematical characters and profound thoughts, which among embodied, are summarized and evaluated objectively. From which the following contributions are obtained:1. Focus on the origin of Weierstrass' Complex Analysis, considering to external and internal causes are the development of complex function in 19^th century reviewed, and three basic approaches lead to complex analysis, namely, algebraic analysis, integral and geometry are introduced. It has cast light on the influences under combinatory analysis in Germany, series work of Gudermann and requirements of rigorous analysis which constitute his motives to develop complex function.2. Overview of extraordinary life of Weierstrass, from life track, academic career as well as pedagogical activities the achievements in different mathematical areas, thought and pedagogical briefly surveyed, which embody his powerful influences on the whole mathematical world in 19^th century.3. Detailed exploration of Weierstrass' three early papers, from the theoretical discussion about integral and series expression along with differential form of analytic function, Weierstrass has independently derived important results such as double series theorem, Cauchy integral theorem and Laurent series theorem etc, which represent the emergence of Weiertrass' methods and the theoretic foundation of complex analysis.4. Research on the theory of analytic factor in his middle period, which reflects the coherence in weierstrass' mathematical thought. What he concerned indicate his algebraic approach in the research. Association with complex function in this study, mathematical thought in this phase inspires the later integer theory of analytic function. 5. Further investigation on the papers which Weierstrass has in late academic period submitted to German Academy of Science, namely during the lectures in Berlin university, the generalization of complex function theory recurring to the properties of analytic function, which shows that the theory of complex function was taken as an independent theory. It has revealed that the theory of complex function has been continuously deepened and the framework of complex function has been formed in this phase.6. Comprehensively analysis of the notes of Weierstrass' students on the lecture of "Introduction to analytic function", function theoretic system is clearly reconstructed. Weierstrass took the concept of "analytic Gebilde" as basic constitution to analytic continuation in order to access to integer analytic function by local ones. Mathematical thoughts, methods and theories of Weierstrass' complex function are extensively analyzed7. Discussion on the tension and limit of impact of Weierstrass' thought on complex function. The research on integral function and meromorphic function oriented to three branches and profoundly guided to different function fields from the end of 19^th to beginning of 20^th century.