difference between real analysis and the complex analysis,the dependence of integral on the path. 2.教学难点 Multi-valuedness of a complex function and the associated branch cut,the path dependence of a complex integral. 3.教学内容 A.definition of a complex function:map from complex number to the complex number:definition of an analytic function. B.Mutli-valuedness of some complex functions,and the introduction of the branch cut. C.continuity and differentiability of a complex function,Cauchy-Riemann condition,and the analytical function D.Integrals of a complex function and its path dependent 4.教学方法 Lectures,discussions,ete 5.教学评价 Corresponding exercises and quiz. 第三章Power Series 1.教学目标 Understand the convergence of a series.and apply several methods to check the 2.教学难点 Convergence vs uniform convergence,Laurent expansion 3.教学内容 A.definition of convergence and uniform convergence. B.the general method to check the convergence of a series C.various methods to obtain the Laurent expansion 4.教学方法 Lectures,discussions,etc 5.教学评价 difference between real analysis and the complex analysis, the dependence of integral on the path. 2. 教学难点 Multi-valuedness of a complex function and the associated branch cut, the path dependence of a complex integral. 3. 教学内容 A. definition of a complex function: map from complex number to the complex number; definition of an analytic function. B. Mutli-valuedness of some complex functions, and the introduction of the branch cut. C. continuity and differentiability of a complex function, Cauchy-Riemann condition, and the analytical function D. Integrals of a complex function and its path dependent 4. 教学方法 Lectures, discussions, etc. 5. 教学评价 Corresponding exercises and quiz. 第三章 Power Series 1. 教学目标 Understand the convergence of a series, and apply several methods to check the convergence. Uniform convergence, convergence circle, local expansion of an analytical function, Taylor and Laurent series 2. 教学难点 Convergence vs uniform convergence, Laurent expansion 3. 教学内容 A. definition of convergence and uniform convergence. B. the general method to check the convergence of a series C. various methods to obtain the Laurent expansion 4.教学方法 Lectures, discussions, etc. 5.教学评价