Integration And Differential Theoryand Series Theory In P-adic Fields

The thesis focuses on p-adic analysis which is mainly discussing the combination of algebras and analysis, a field specializing in analysis.The thesis tell us that:1. It introduces the general situation of p-adic's. In recent years, manycountries scholars get works on p-adic analysis.2. It introduces fundamental knowledges, preparing for continues working.3. The works that I was doing is three parts as follows:1) .It defines the integration of variable p-adic and discusses the necessary and sufficient condition with complete and detailed demonstration.2) Mt defines the concept of p-adic variable real-valued function and integralmean-value theorem via the analysis of the relationship between integration and differential.3) .It analyzes the convergence of p-adic infinite series and discusses conceptof p p - adic series of constant terms as well as tits necessary and sufficient condition with complete and detailed demonstration. It makes the conclusion that it is much easier to estimate series astringency in p - adic field than in real number field.In all, the thesis is to lay a sound basis for further research and application of p-adic.