东北大学：某学院应用通信工程专业《复变函数与积分变换》课程教学大纲

复变函数与积分变换教学大纲Complex Variable FunctionSubject Syllabusand Integral Transformation一、课程信息SubjectInformation课程编号：开课学期：33100111001Subject IDSemester课程分类：所属课群：公共基础 GF专业基础MFSectionCategory课程学分：总学时/周：232/8Credit PointsTotal Hours/Weeks理论学时：实验学时：320EXP. HoursLECT.HoursPBL学时：实践学时/周：00PBL HoursPRAC. Hours/Weeks开课学院：东北大学适用专业：通信工程CECollegeStream悉尼智能科技学院课程属性：课程模式：必修Compulsory自建NEUModePattern中方课程协调人刘立卿成绩记载方式：百分制MarksNEU CoordinatorLiu LiqingResult Type高等数学建模（一），高等数学建模（二)先修课程Advancedmathematical modeling ()RequisitesAdvancedmathematicalmodeling(Il)JamesWardBrown,ComplexVariables andApplications(7thedition)英文参考教材机械工业出版社ENTextbooksYunying Gai, Yuming Xing, Functions of a Complex Variable andInteralTransforms，科学出版社，2007西安交通大学高等数学教研室编：《复变函数》（第四版），高等中文参考教材教育出版社，1996CN Textbooks张元林：《积分变换》（第六版），高等教育出版社，2019教学资源：https://www.08nm.com/c38.htmlResources课程负责人(撰写人)：提交日期：单击或点击此处输Subject DirectorSubmitted Date入日期。刘立卿任课教师(含负责人)：Liqing LiuTaught by审核人：批准人：韩鹏史闻博Checked byApproved by批准日期：单击或点击此处输ApprovedDate入日期。1 / 14

1 / 14 复变函数与积分变换 教学大纲 Subject Syllabus Complex Variable Function and Integral Transformation 一、课程信息 Subject Information 课程编号: Subject ID 3100111001 开课学期: Semester 3 课程分类: Category 公共基础 GF 所属课群: Section 专业基础 MF 课程学分: Credit Points 2 总学时/周: Total Hours/Weeks 32/8 理论学时: LECT. Hours 32 实验学时: EXP. Hours 0 PBL 学时: PBL Hours 0 实践学时/周: PRAC. Hours/Weeks 0 开课学院: College 东北大学 悉尼智能科技学院 适用专业: Stream 通信工程 CE 课程属性: Pattern 必修 Compulsory 课程模式: Mode 自建 NEU 中方课程协调人: NEU Coordinator 刘立卿 Liu Liqing 成绩记载方式: Result Type 百分制 Marks 先修课程: Requisites 高等数学建模（一），高等数学建模（二） Advanced mathematical modeling (I), Advanced mathematical modeling (II) 英文参考教材: EN Textbooks James Ward Brown, Complex Variables and Applications(7th edition) 机械工业出版社 Yunying Gai, Yuming Xing, Functions of a Complex Variable and Interal Transforms, 科学出版社，2007 中文参考教材: CN Textbooks 西安交通大学高等数学教研室编：《复变函数》（第四版），高等 教育出版社，1996 张元林：《积分变换》（第六版），高等教育出版社，2019 教学资源: Resources https://www.08nm.com/c_38.html 课程负责人(撰写人): Subject Director 提交日期: Submitted Date 单击或点击此处输 入日期。 任课教师(含负责人): Taught by 刘立卿 Liqing Liu 审核人: Checked by 韩鹏 批准人: Approved by 史闻博 批准日期: Approved Date 单击或点击此处输 入日期

二、教学目标SubjectLearningObjectives（SLOs）注：毕业要求及指标点可参照悉尼学院本科生培养方案，可根据实际情况增减行数Note: GA and index can be referred from undergraduate program in SSTC website. Please add/reduce lines based on subject.复变函数与积分变换是运用复变函数的理论知识解决微分方程和积分方程等实际问题的一门课程，同时是通信工程专业必修的数学基础课。其知识内容主要包括复变函数的定义、连续性、解析函数的概念与判定准则、几个初等函数、复变函数的积分、级数、留数、Fourier变换、Laplace变换等方面的内容。复变函数与积分变换既能简化计算，又能体现明确的物理意义，在许多领域有广泛应用，如电气工程、通信与控制、信号分析与图像处理、机械系统、流体力学、地质勘探与地震预报等工程技术领域。通过本课程的学习，学生不仅可以掌握复变函数与积分变换的基础理论及工程技术中的常用数学方法，为后续有关课程的学习奠定了必要的数学基础，而且培养学生抽象思维能力、逻辑推理能力、空间想象能力和科学计算等能力，培养学生勤学笃行意识和创新实践精神，厚植学生四个自信和家国情怀。Complex variablefunction and integral transformation is a courseto solve practical problems such as differential equations and integralequationsby usingthe theoretical knowledge of complex variablefunction.Itis also a compulsory basic mathematics course for整体目标：communication engineering majors.Its knowledge content mainlyOverallObjectiveincludes thedefinition ofcomplex variable function,continuity,theconcept and judgment criteria of analytical function, several elementaryfunctions,integral,series,residue,Fouriertransform,Laplacetransformand so on.Complexvariablefunction and integral transformation cannot only simplify calculation, but also reflect clear physicalsignificance. They are widely used in many fields, such as electricalengineering, communication and control, signal analysis and imageprocessing, mechanical system, hydrodynamics, geological explorationand earthquake prediction,Through the study of this course, studentscan not onlymasterthe basictheory of complexvariablefunction andintegral transformation and the common mathematical methods inengineering technology,lay a necessary mathematical foundationforthe study of subsequent relevant courses, but also cultivate studentsabstract thinking ability,logical reasoning ability, spatial imaginationabilityand scientific calculationability,Cultivate students'awarenessofdiligent study and practice and innovative practice spirit, and cultivatestudents'four self-confidence and feelings of family and country.了解区域与复变函数的概念。Recognizetheconcepts of region and complex variablefunction.（1）专业目标：1-1理解复变函数的极限和连续的概念。Professional AbilityUnderstand the concept of limit and continuity of complexvariablefunction掌握复数的各种表示方法及其运算。2/14

2 / 14 二、教学目标 Subject Learning Objectives (SLOs) 注：毕业要求及指标点可参照悉尼学院本科生培养方案，可根据实际情况增减行数 Note: GA and index can be referred from undergraduate program in SSTC website. Please add/reduce lines based on subject. 整体目标: Overall Objective 复变函数与积分变换是运用复变函数的理论知识解决微分方 程和积分方程等实际问题的一门课程，同时是通信工程专业必修的 数学基础课。其知识内容主要包括复变函数的定义、连续性、解析 函数的概念与判定准则、几个初等函数、复变函数的积分、级数、 留数、Fourier 变换、Laplace 变换等方面的内容。复变函数与积分 变换既能简化计算，又能体现明确的物理意义，在许多领域有广泛 应用，如电气工程、通信与控制、信号分析与图像处理、机械系统、 流体力学、地质勘探与地震预报等工程技术领域。通过本课程的学 习，学生不仅可以掌握复变函数与积分变换的基础理论及工程技术 中的常用数学方法，为后续有关课程的学习奠定了必要的数学基 础，而且培养学生抽象思维能力、逻辑推理能力、空间想象能力和 科学计算等能力，培养学生勤学笃行意识和创新实践精神，厚植学 生四个自信和家国情怀。 Complex variable function and integral transformation is a course to solve practical problems such as differential equations and integral equations by using the theoretical knowledge of complex variable function. It is also a compulsory basic mathematics course for communication engineering majors. Its knowledge content mainly includes the definition of complex variable function, continuity, the concept and judgment criteria of analytical function, several elementary functions, integral, series, residue, Fourier transform, Laplace transform and so on. Complex variable function and integral transformation can not only simplify calculation, but also reflect clear physical significance. They are widely used in many fields, such as electrical engineering, communication and control, signal analysis and image processing, mechanical system, hydrodynamics, geological exploration and earthquake prediction. Through the study of this course, students can not only master the basic theory of complex variable function and integral transformation and the common mathematical methods in engineering technology, lay a necessary mathematical foundation for the study of subsequent relevant courses, but also cultivate students' abstract thinking ability, logical reasoning ability, spatial imagination ability and scientific calculation ability, Cultivate students' awareness of diligent study and practice and innovative practice spirit, and cultivate students' four self-confidence and feelings of family and country. （1）专业目标: Professional Ability 1-1 了解区域与复变函数的概念。 Recognize the concepts of region and complex variable function. 理解复变函数的极限和连续的概念。 Understand the concept of limit and continuity of complex variable function. 掌握复数的各种表示方法及其运算

Master various representationmethods and operations ofcomplexnumbers了解复变函数的导数及复变函数解析的概念；了解调和函数与解析函数的关系。Recognizethederivativeof complexvariablefunctionand theconceptof complexvariablefunction analysis;Recognizetherelationshipbetweenharmonicfunctionandanalyticalfunction理解指数函数、三角函数、对数函数及幂函数的定义及它们1-2的主要性质（包括在单值区域中的解析性）。Understand thecomplex function,understand thedefinitions ofexponential function,trigonometric function,logarithmicfunction and power function and their main properties(including analyticity in single valued region)掌握复变函数解析的充要条件：从解析函数的实（虚）部求其虚（实）部的方法。了解闭路变形原理。Recognize theprinciple of closed-circuit deformation理解复变函数积分的定义，解析函数无限次可导的性质。Understand the definition of complex variable function integraland analyze the infinite derivative of function.1-3掌握柯西一古萨基本定理；复合闭路定理，柯西积分定理，和高阶导数公式，解析函数与调和函数的关系。MasterthebasicCauchykusAtheorem,Compoundclosed-circuittheorem,Cauchy integral theorem, andhigher-order derivative formula, the relationship betweenanalytical functionand harmonicfunction了解复数项级数收敛、发散及绝对收敛等概念；幂级数收敛圆的概念。Recognize the concepts of series convergence, divergence andabsolute convergence of complex terms;The conceptof powerseries convergence circle理解幂级数在收敛圆内一些基本性质：简单的幂级数收敛半径的求法。dsomebasicpropertiesUnderstandsof powerseriesinconvergent circle;A simplemethod for finding the1-4convergenceradius ofpowerseries.掌握将解析函数在一点展开为泰勒级数，熟记的麦克劳林展开式，并能利用它们将一些简单的解析函数展开为幂级数；掌握将函数在圆环域展开为洛朗级数的间接法。Master the expansion of analytical functions into Taylor seriesat one point and the familiar McLaughlin expansion, and canuse them to expand some simple analytical functions intopower series; Master the indirect method of expanding thefunction into Laurent series in the ring domain3 / 14

3 / 14 Master various representation methods and operations of complex numbers. 1-2 了解复变函数的导数及复变函数解析的概念；了解调和函数 与解析函数的关系。 Recognize the derivative of complex variable function and the concept of complex variable function analysis; Recognize the relationship between harmonic function and analytical function. 理解指数函数、三角函数、对数函数及幂函数的定义及它们 的主要性质（包括在单值区域中的解析性）。 Understand the complex function, understand the definitions of exponential function, trigonometric function, logarithmic function and power function and their main properties (including analyticity in single valued region). 掌握复变函数解析的充要条件；从解析函数的实（虚）部求 其虚（实）部的方法。 1-3 了解闭路变形原理。 Recognize the principle of closed-circuit deformation. 理解复变函数积分的定义，解析函数无限次可导的性质。 Understand the definition of complex variable function integral and analyze the infinite derivative of function. 掌握柯西—古萨基本定理；复合闭路定理，柯西积分定理， 和高阶导数公式，解析函数与调和函数的关系。 Master the basic Cauchy kusA theorem; Compound closed-circuit theorem, Cauchy integral theorem, and higher-order derivative formula, the relationship between analytical function and harmonic function. 1-4 了解复数项级数收敛、发散及绝对收敛等概念；幂级数收敛 圆的概念。 Recognize the concepts of series convergence, divergence and absolute convergence of complex terms; The concept of power series convergence circle. 理解幂级数在收敛圆内一些基本性质；简单的幂级数收敛半 径的求法。 Understand some basic properties of power series in convergent circle; A simple method for finding the convergence radius of power series. 掌握将解析函数在一点展开为泰勒级数，熟记的麦克劳林展 开式，并能利用它们将一些简单的解析函数展开为幂级数； 掌握将函数在圆环域展开为洛朗级数的间接法。 Master the expansion of analytical functions into Taylor series at one point and the familiar McLaughlin expansion, and can use them to expand some simple analytical functions into power series; Master the indirect method of expanding the function into Laurent series in the ring domain

了解函数在无穷远点的性态，会用留数求一些实积分。Recognizethe behavior of thefunction at infinity,and beableto find some real integrals with residues.理解留数的概念。Understand the concept of residue1-5掌握判断孤立奇点的类型，极点处留数的求法；留数定理；用留数求闭曲线积分的方法。Master the method of judging the type of isolated singularityand finding the residue at the pole; Residue theorem, Themethod of calculating closed curve integral with residue了解Fourier积分。Understand Fourier integral理解单位脉冲函数，卷积和卷积定理。Understand unit pulse function, convolution and convolutiontheorem1-6掌握Fourier变换的概念：掌握Fourier变换的线性性质、位移性质、微分性质、积分性质。Master the concept of Fourier transform; Master the linear,displacement, differential and integral properties of Fouriertransform了解Laplace变换的概念。Understandtheconceptof Laplacetransform理解Laplace逆变换定理(海维赛德展开式)，Laplace变换的卷积定理，Laplace变换的线性性质、位移性质、微分性质、积分性质。Understandtheconceptof Laplacetransform.Understandtheinverse Laplace transform theorem (haverside expansion), theconvolution theorem of Laplace transform, and the linear,displacement,differential and integral properties of Laplacetransform.1-7掌握某些常见函数(例如指数函数、sinkt,coskt,S(t的变换公式,并会查表求象函数和象原函数.会用Laplace变换解常系数线性微分方程及方程组。Master the transformation formulas of some common functions(such as exponential function andsin kt cos kt (t), andbe able to look up the table to find the image function andimage primitive function.Be able to use Laplacetransformation to solve linear differential equations andequationswith constant coefficients理解复变分析知识对于刻画工程实践问题的重要意义。2-1（2）德育目标：Understandthe significant meanings of thecomplex analysis inEssential Qualitydepictingthepractical engineeringproblems2-2让学生通过学习，掌握事物发展规律，通晓天下道理，丰富4 / 14

4 / 14 1-5 了解函数在无穷远点的性态，会用留数求一些实积分。 Recognize the behavior of the function at infinity, and be able to find some real integrals with residues. 理解留数的概念。 Understand the concept of residue. 掌握判断孤立奇点的类型，极点处留数的求法；留数定理； 用留数求闭曲线积分的方法。 Master the method of judging the type of isolated singularity and finding the residue at the pole; Residue theorem; The method of calculating closed curve integral with residue. 1-6 了解 Fourier 积分。 Understand Fourier integral. 理解单位脉冲函数，卷积和卷积定理。 Understand unit pulse function, convolution and convolution theorem. 掌握 Fourier 变换的概念；掌握 Fourier 变换的线性性质﹑位 移性质﹑微分性质﹑积分性质。 Master the concept of Fourier transform; Master the linear, displacement, differential and integral properties of Fourier transform. 1-7 了解 Laplace 变换的概念。 Understand the concept of Laplace transform. 理解 Laplace 逆变换定理(海维赛德展开式), Laplace 变换的 卷积定理, Laplace 变换的线性性质﹑位移性质﹑微分性质 ﹑积分性质。 Understand the concept of Laplace transform. Understand the inverse Laplace transform theorem (haverside expansion), the convolution theorem of Laplace transform, and the linear, displacement, differential and integral properties of Laplace transform. 掌握某些常见函数(例如指数函数﹑ sin kt , cos kt , t 的变换公式,并会查表求象函数和象原函数.会用 Laplace 变 换解常系数线性微分方程及方程组。 Master the transformation formulas of some common functions (such as exponential function and sin kt cos kt , t ), and be able to look up the table to find the image function and image primitive function. Be able to use Laplace transformation to solve linear differential equations and equations with constant coefficients. （2）德育目标: Essential Quality 2-1 理解复变分析知识对于刻画工程实践问题的重要意义。 Understand the significant meanings of the complex analysis in depicting the practical engineering problems. 2-2 让学生通过学习，掌握事物发展规律，通晓天下道理，丰富

学识，增长见识，塑造品格，努力成为德智体美劳全面发展的社会主义建设者和接班人。Let students master the law of development of things,understand the truth of the world, enrich their knowledgeincrease their knowledge, shape their character, and strive tobecome socialist builders and successors with all-rounddevelopment of morality, intelligence, physique, beauty andlabor.展示本专业在新时代中国特色社会主义建设中的成就和当前要解决的重大课题。2-3Displaytheachievementsof thismajorintheconstructionofsocialism with Chinesecharacteristics in the new era and themajor issuestobe solved atpresent注重科学思维方法的训练和科学伦理的教育，培养学生探索未知、追求真理、勇攀科学高峰的责任感和使命感。Pay attention to the training of scientific thinking methods and2-4the education of scientific ethics, and cultivate students'senseof responsibilityand mission to explore the unknown, pursuethe truth and climb thepeak of science.课程教学目标与毕业要求的对应关系MatrixofGA&SLOs毕业要求GA指标点GAIndex教学目标SLOs1、工程知识：能够将数学、指标点1-1：掌握数学、自然科学、工程自然科学、工程基础和专业基础和专业知识，并使用其建立正确的1-1 到 1-7知识用于解决复杂工程问数学、物理学等模型以解释复杂工程问2-1,2-2题。题；指标点2-1：能够应用数学、自然科学和工程科学的基本原理、方法和手段，分2、问题分析：能够应用数学、析、识别、表达本专业相关的复杂工程自然科学和工程科学的基本问题；原理、方法和手段，识别、1-1 到 1-7指标点2-2：能够应用数学、自然科学和表达、并通过文献研究分析2-2,2-3工程科学的基本原理、方法和手段，针复杂工程问题，以获得有效对实际复杂工程问题设计针对性的技术结论。方案，并综合运用文献、科学理论和技术手段予以解决。4、研究：能够基于科学原理并采用科学方法对复杂工程指标点4-2：能够结合本专业知识对实验问题进行研究，包括设计实数据进行分析与解释，设计并优化实验1-1到1-7验、分析与解释数据、并通方案，并通过信息综合得到合理有效的2-2, 2-3, 2-4过信息综合得到合理有效的结论。结论。三、教学内容Content(Topics)注：以中英文填写，各部分内容的表格可根据实际知识单元数量进行复制、扩展或缩减Note: Filled in both CN and EN, extend or reduce based on the actual numbers of knowledge uni5 / 14

5 / 14 学识，增长见识，塑造品格，努力成为德智体美劳全面发展 的社会主义建设者和接班人。 Let students master the law of development of things, understand the truth of the world, enrich their knowledge, increase their knowledge, shape their character, and strive to become socialist builders and successors with all-round development of morality, intelligence, physique, beauty and labor. 2-3 展示本专业在新时代中国特色社会主义建设中的成就和当 前要解决的重大课题。 Display the achievements of this major in the construction of socialism with Chinese characteristics in the new era and the major issues to be solved at present. 2-4 注重科学思维方法的训练和科学伦理的教育，培养学生探索 未知、追求真理、勇攀科学高峰的责任感和使命感。 Pay attention to the training of scientific thinking methods and the education of scientific ethics, and cultivate students' sense of responsibility and mission to explore the unknown, pursue the truth and climb the peak of science. 课程教学目标与毕业要求的对应关系 Matrix of GA & SLOs 毕业要求 GA 指标点 GA Index 教学目标 SLOs 1、工程知识：能够将数学、 自然科学、工程基础和专业 知识用于解决复杂工程问 题。 指标点 1-1：掌握数学、自然科学、工程 基础和专业知识，并使用其建立正确的 数学、物理学等模型以解释复杂工程问 题； 1-1 到 1-7 2-1,2-2 2、问题分析：能够应用数学、 自然科学和工程科学的基本 原理、方法和手段，识别、 表达、并通过文献研究分析 复杂工程问题，以获得有效 结论。 指标点 2-1：能够应用数学、自然科学和 工程科学的基本原理、方法和手段，分 析、识别、表达本专业相关的复杂工程 问题； 1-1 到 1-7 2-2,2-3 指标点 2-2：能够应用数学、自然科学和 工程科学的基本原理、方法和手段，针 对实际复杂工程问题设计针对性的技术 方案，并综合运用文献、科学理论和技 术手段予以解决。 4、研究：能够基于科学原理 并采用科学方法对复杂工程 问题进行研究，包括设计实 验、分析与解释数据、并通 过信息综合得到合理有效的 结论。 指标点 4-2：能够结合本专业知识对实验 数据进行分析与解释，设计并优化实验 方案，并通过信息综合得到合理有效的 结论。 1-1 到 1-7 2-2, 2-3, 2-4 三、教学内容 Content (Topics) 注：以中英文填写，各部分内容的表格可根据实际知识单元数量进行复制、扩展或缩减 Note: Filled in both CN and EN, extend or reduce based on the actual numbers of knowledge unit

(1)理论教学 Lecture知识单元序号：支撑教学目标：11-1，2-1，2-2Knowledge Unit No.SLOs Supported知识单元名称复数与复变函数ComplexandcomplexvariablefunctionUnit Title复数及其代数运算与几何表示Complex and its algebraic operation and geometric representation复数的乘幂与方根Powerandsquarerootofcomplex知识点区域TheregionKnowledge Delivery复变函数Complexvariablefunction复变函数的极限和连续性Limitationandcontinuityofcomplexvariablefunction了解：区域与复变函数的概念RecognizeThe concept of region and complex variable function复变函数的极限和连续的概念学习目标：理解：The conceptof limitation and continuityof complexUnderstandLearning Objectivesvariable function掌握：复数的各种表示方法及其运算MasterVariousexpressions andoperations of complex2-1理解复变分析知识对于刻画工程实践问题的重要意义。Understand the significant meanings of the complex analysis indepicting the practical engineering problems.2-2让学生通过学习，掌握事物发展规律，通晓天下道理，丰富学识，增长见识，塑造品格，努力成为德智体美劳全面发展的社会主德育目标义建设者和接班人。Moral ObjectivesLet students master the law of development of things, understand thetruth of the world, enrich their knowledge, increase their knowledge,shape their character, and strive to become socialist builders andsuccessors with all-round development of morality, intelligence,physique, beauty and labor.复数的各种表示方法及相应的运算；区域与连通性：复变函数的极重点：限 Various representation methods and corresponding operations ofKey Pointscomplex,theregion and connectivity;the limitation of complexfunction难点：扩充复球面；复数开方和多值函数Focal pointsExtendedcomplexsphere;Complexrootandmultivaluedfunction知识单元序号：支撑教学目标：21-2，2-1,2-3Knowledge Unit No.SLOs Supported知识单元名称解析函数AnalyticfunctionUnit Title复变函数的导数解析函数的概念知识点：Derivativeofcomplexfunction,conceptofanalyticfunctionKnowledge Delivery函数解析的充要条件6/14

6 / 14 (1) 理论教学 Lecture 知识单元序号: Knowledge Unit No. 1 支撑教学目标: SLOs Supported 1-1，2-1，2-2 知识单元名称 Unit Title 复数与复变函数 Complex and complex variable function 知识点: Knowledge Delivery 复数及其代数运算与几何表示 Complex and its algebraic operation and geometric representation 复数的乘幂与方根 Power and square root of complex 区域 The region 复变函数 Complex variable function 复变函数的极限和连续性 Limitation and continuity of complex variable function 学习目标: Learning Objectives 了解: Recognize 区域与复变函数的概念 The concept of region and complex variable function 理解: Understand 复变函数的极限和连续的概念 The concept of limitation and continuity of complex variable function 掌握: Master 复数的各种表示方法及其运算 Various expressions and operations of complex 德育目标 Moral Objectives 2-1 理解复变分析知识对于刻画工程实践问题的重要意义。 Understand the significant meanings of the complex analysis in depicting the practical engineering problems. 2-2 让学生通过学习，掌握事物发展规律，通晓天下道理，丰富学 识，增长见识，塑造品格，努力成为德智体美劳全面发展的社会主 义建设者和接班人。 Let students master the law of development of things, understand the truth of the world, enrich their knowledge, increase their knowledge, shape their character, and strive to become socialist builders and successors with all-round development of morality, intelligence, physique, beauty and labor. 重点: Key Points 复数的各种表示方法及相应的运算；区域与连通性；复变函数的极 限 Various representation methods and corresponding operations of complex; the region and connectivity; the limitation of complex function 难点: Focal points 扩充复球面；复数开方和多值函数 Extended complex sphere; Complex root and multi valued function 知识单元序号: Knowledge Unit No. 2 支撑教学目标: SLOs Supported 1-2，2-1，2-3 知识单元名称 Unit Title 解析函数 Analytic function 知识点: Knowledge Delivery 复变函数的导数,解析函数的概念 Derivative of complex function, concept of analytic function 函数解析的充要条件

Necessaryand sufficientconditionsforanalyticfunction解析函数与调和函数的关系Therelationshipbetween analyticfunction and harmonicfunction初等函数：指数函数，对数函数，乘幂与幂函数：三角函数Elementaryfunction:exponential function,logarithmicfunction,powerand power function; trigonometric function复变函数的导数及复变函数解析的概念了解：Thederivative of complex variable function and theRecognizeconceptof complexvariablefunctionanalysis调和函数与解析函数的关系理解：The relationship between analyticfunctionandUnderstand学习目标：harmonicfunctionLearning Objectives复变函数解析的充要条件，指数函数、三角函数、对数函数及幂函数的定义及它们的主要性质掌握：Necessaryand sufficient conditionsforanalyticMasterfunction,the concept and nature of exponentialfunction,logarithmicfunction,powerandpowerfunction; trigonometric function2-1理解复变分析知识对于刻画工程实践问题的重要意义。Understand the significant meanings of the complex analysis indepictingthepracticalengineeringproblems德育目标2-3展示本专业在新时代中国特色社会主义建设中的成就和当前要Moral Objectives解决的重大课题。Display the achievements of this major in the construction of socialismwith Chinese characteristics in thenew era and themajor issues tobesolved at present复变函数的导数；函数解析的充要条件；解析函数与调和函数的关系：初等函数重点：Property the derivative of complex variable function; The necessaryKey Pointsand sufficient conditions of function analysis:Therelation betweenanalyticfunctionand harmonicfunction;Elementaryfunction难点：解析函数的判别Focal pointsDiscrimination of analytic functions知识单元序号：支撑教学目标31-3,2-2,2-4Knowledge Unit No.SLOs Supported知识单元名称复变函数积分IntegralofcomplexfunctionUnit Title复变函数积分的概念Theconceptofcomplexfunctionintegral柯西一古萨基本定理：Cauchy-Gusa basictheorem基本定理的推广一复合闭路定理知识点：Generalization of the fundamental theorem compound closed circuitKnowledge Deliverytheorem柯西积分公式：Cauchyintegralformula解析函数的高阶导数Higher order derivatives of analytic functions7/14

7 / 14 Necessary and sufficient conditions for analytic function 解析函数与调和函数的关系 The relationship between analytic function and harmonic function 初等函数:指数函数,对数函数,乘幂 与幂函数；三角函数 Elementary function: exponential function, logarithmic function, power and power function; trigonometric function 学习目标: Learning Objectives 了解: Recognize 复变函数的导数及复变函数解析的概念 The derivative of complex variable function and the concept of complex variable function analysis 理解: Understand 调和函数与解析函数的关系 The relationship between analytic function and harmonic function 掌握: Master 复变函数解析的充要条件，指数函数、三角函数、对 数函数及幂函数的定义及它们的主要性质 Necessary and sufficient conditions for analytic function, the concept and nature of exponential function, logarithmic function, power and power function; trigonometric function 德育目标 Moral Objectives 2-1 理解复变分析知识对于刻画工程实践问题的重要意义。 Understand the significant meanings of the complex analysis in depicting the practical engineering problems. 2-3 展示本专业在新时代中国特色社会主义建设中的成就和当前要 解决的重大课题。 Display the achievements of this major in the construction of socialism with Chinese characteristics in the new era and the major issues to be solved at present. 重点: Key Points 复变函数的导数；函数解析的充要条件；解析函数与调和函数的关 系；初等函数 Property the derivative of complex variable function; The necessary and sufficient conditions of function analysis; The relation between analytic function and harmonic function; Elementary function 难点: Focal points 解析函数的判别 Discrimination of analytic functions 知识单元序号: Knowledge Unit No. 3 支撑教学目标: SLOs Supported 1-3，2-2，2-4 知识单元名称 Unit Title 复变函数积分 Integral of complex function 知识点: Knowledge Delivery 复变函数积分的概念 The concept of complex function integral 柯西—古萨基本定理：Cauchy-Gusa basic theorem 基本定理的推广—复合闭路定理 Generalization of the fundamental theorem compound closed circuit theorem 柯西积分公式：Cauchy integral formula 解析函数的高阶导数 Higher order derivatives of analytic functions

解析函数与调和函数的关系The relationship between analytic function and harmonic function复变函数积分的性质，解析函数无限次可导的性质了解：Thepropertiesof theintegralof thefunctionofcomplexRecognizevariableandthepropertiesoftheinfinitelydifferentiable analyticfunction高阶导数公式，解析函数无限次可导的性质理解：学习目标：Higher order derivativeformula,theproperty of infiniteUnderstandLearning Objectivesderivative of analytic function复变函数积分的定义，柯西积分定理，会求复变函数的积分掌握：The definition offunction integral of complex variable, MasterCauchy integral theorem, can find the integral offunctionofcomplexvariable2-2让学生通过学习，掌握事物发展规律，通晓天下道理，丰富学识，增长见识，塑造品格，努力成为德智体美劳全面发展的社会主义建设者和接班人。Let students masterthe law of development of things, understand thetruth of the world, enrich their knowledge, increase their knowledge,shape their character, and strive to become socialist builders and德育目标successors with all-round development of morality,intelligenceMoral Objectivesphysique, beauty and labor2-4注重科学思维方法的训练和科学伦理的教育，培养学生探索未知、追求真理、勇攀科学高峰的责任感和使命感。Pay attention to the training of scientific thinking methods and theeducation of scientific ethics, and cultivate students' sense ofresponsibility and missiontoexploretheunknown,pursuethetruth andclimbthepeakofscience.柯西一古萨基本定理；复合闭路定理；柯西积分公式；高阶导数公重点：式 Cauchy-Gusa basic theorem; Compound closed circuit theorem;Key PointsCauchy integral formula; Higher order derivative formula复变函数积分的概念：多连通区域上的柯西积分定理；解析函数的难点：无穷可微性TheconceptoffunctionintegralofcomplexvariableFocal pointsCauchyintegral theoremon multiconnected domain, Infinitedifferentiability of analytic functions知识单元序号：支撑教学目标：41-4, 2-3, 2-4SLOs SupportedKnowledge Unit No知识单元名称级数SeriesUnit Title复数项级数Seriesofcomplexterms知识点：幂级数PowerseriesrKnowledge Delivery泰勒级数Taylorseries洛朗级数Laurent series学习目标：了解：复数项级数收敛、发散及绝对收敛等概念，幂级数收8/14

8 / 14 解析函数与调和函数的关系 The relationship between analytic function and harmonic function 学习目标: Learning Objectives 了解: Recognize 复变函数积分的性质，解析函数无限次可导的性质 The properties of the integral of the function of complex variable and the properties of the infinitely differentiable analytic function 理解: Understand 高阶导数公式，解析函数无限次可导的性质 Higher order derivative formula, the property of infinite derivative of analytic function 掌握: Master 复变函数积分的定义，柯西积分定理，会求复变函数 的积分 The definition of function integral of complex variable, Cauchy integral theorem, can find the integral of function of complex variable 德育目标 Moral Objectives 2-2 让学生通过学习，掌握事物发展规律，通晓天下道理，丰富学 识，增长见识，塑造品格，努力成为德智体美劳全面发展的社会主 义建设者和接班人。 Let students master the law of development of things, understand the truth of the world, enrich their knowledge, increase their knowledge, shape their character, and strive to become socialist builders and successors with all-round development of morality, intelligence, physique, beauty and labor. 2-4 注重科学思维方法的训练和科学伦理的教育，培养学生探索未 知、追求真理、勇攀科学高峰的责任感和使命感。 Pay attention to the training of scientific thinking methods and the education of scientific ethics, and cultivate students' sense of responsibility and mission to explore the unknown, pursue the truth and climb the peak of science. 重点: Key Points 柯西—古萨基本定理；复合闭路定理；柯西积分公式；高阶导数公 式 Cauchy-Gusa basic theorem; Compound closed circuit theorem; Cauchy integral formula; Higher order derivative formula 难点: Focal points 复变函数积分的概念；多连通区域上的柯西积分定理；解析函数的 无穷可微性 The concept of function integral of complex variable; Cauchy integral theorem on multi connected domain; Infinite differentiability of analytic functions 知识单元序号: Knowledge Unit No. 4 支撑教学目标: SLOs Supported 1-4，2-3，2-4 知识单元名称 Unit Title 级数 Series 知识点: Knowledge Delivery 复数项级数 Series of complex terms 幂级数 Power series r 泰勒级数 Taylor series 洛朗级数 Laurent series 学习目标: 了解: 复数项级数收敛、发散及绝对收敛等概念，幂级数收

敛圆的概念Theconceptofconvergence,divergenceLearning ObjectivesRecognizeand absolute convergence of complex series, and theconceptofconvergencecircleofpowerseries简单的函数在其孤立奇点附近展开为罗朗级数的间理解：接法UnderstandAn indirect method to expand a simple function into aLaurent series near its isolated singular point简单的幂级数收敛半径的求法，一些函数的麦克劳林掌握：展开式Simplepower seriesconvergenceradiusoftheMastersolution,somefunctionsofMcLaughlinexpansion2-3展示本专业在新时代中国特色社会主义建设中的成就和当前要解决的重大课题。Displaythe achievements ofthis major in the construction of socialismwith Chinese characteristics in the new era and the major issues to besolved at present.德育目标2-4注重科学思维方法的训练和科学伦理的教育，培养学生探索未Moral Objectives知、追求真理、勇攀科学高峰的责任感和使命感。Pay attention to the training of scientific thinking methods and theeducation of scientific ethics, and cultivate students' sense ofresponsibilityand mission toexploretheunknown,pursuethetruth andclimb the peak of science.收敛半径的求法：复变函数展成泰勒级数：复变函数展成洛朗级数重点：The solution of convergence radius;The function of complex variableKey Pointsis expanded into Taylor series; Expansion of complex function intoLaurent series级数在收敛区间端点的敛散性，利用间接方法将函数展开成洛朗级难点：数Focal pointsThe function is expanded into Laurent series by indirect method知识单元序号：支撑教学目标：51-5,2-2,2-3SLOs SupportedKnowledge Unit No知识单元名称留数ResidueUnit Title可去奇点、极点、本性奇点、函数的零点及极点的关系The removable singularities, poles, essential singularities and the知识点：relations among the zeros of functions and polesKnowledge Delivery留数：留数的定义及留数定理，留数的计算规则Residue: the definition of residue and residue theorem, residuecalculation rules了解：留数的概念RecognizeThe concept of residue学习目标：理解：扩充复平面上的留数定理Learning ObjectivesUnderstandResidue theorem on extended complexplane掌握：极点处留数的求法：留数定理；用留数求围道积分的Master方法9/14

9 / 14 Learning Objectives Recognize 敛圆的概念 The concept of convergence, divergence and absolute convergence of complex series, and the concept of convergence circle of power series 理解: Understand 简单的函数在其孤立奇点附近展开为罗朗级数的间 接法 An indirect method to expand a simple function into a Laurent series near its isolated singular point 掌握: Master 简单的幂级数收敛半径的求法，一些函数的麦克劳林 展开式 Simple power series convergence radius of the solution, some functions of McLaughlin expansion 德育目标 Moral Objectives 2-3 展示本专业在新时代中国特色社会主义建设中的成就和当前要 解决的重大课题。 Display the achievements of this major in the construction of socialism with Chinese characteristics in the new era and the major issues to be solved at present. 2-4 注重科学思维方法的训练和科学伦理的教育，培养学生探索未 知、追求真理、勇攀科学高峰的责任感和使命感。 Pay attention to the training of scientific thinking methods and the education of scientific ethics, and cultivate students' sense of responsibility and mission to explore the unknown, pursue the truth and climb the peak of science. 重点: Key Points 收敛半径的求法；复变函数展成泰勒级数；复变函数展成洛朗级数 The solution of convergence radius; The function of complex variable is expanded into Taylor series; Expansion of complex function into Laurent series 难点: Focal points 级数在收敛区间端点的敛散性，利用间接方法将函数展开成洛朗级 数 The function is expanded into Laurent series by indirect method 知识单元序号: Knowledge Unit No. 5 支撑教学目标: SLOs Supported 1-5，2-2，2-3 知识单元名称 Unit Title 留数 Residue 知识点: Knowledge Delivery 可去奇点﹑极点﹑本性奇点、函数的零点及极点的关系 The removable singularities, poles, essential singularities and the relations among the zeros of functions and poles 留数:留数的定义及留数定理,留数的计算规则 Residue: the definition of residue and residue theorem, residue calculation rules 学习目标: Learning Objectives 了解: Recognize 留数的概念 The concept of residue 理解: Understand 扩充复平面上的留数定理 Residue theorem on extended complex plane 掌握: Master 极点处留数的求法；留数定理；用留数求围道积分的 方法

The solution of residue at the pole; Residue theorem, Amethod of finding contour integral byresidue2-2让学生通过学习，掌握事物发展规律，通晓天下道理，丰富学识，增长见识，塑造品格，努力成为德智体美劳全面发展的社会主义建设者和接班人。Let students master the law of development of things, understand thetruth of the world, enrich their knowledge, increase their knowledge,shape their character, and strive to become socialist builders and德育目标successors with all-round development of morality,intelligence,Moral Objectivesphysique, beauty and labor2-3展示本专业在新时代中国特色社会主义建设中的成就和当前要解决的重大课题。Displaytheachievementsof thismajorintheconstructionof socialismwith Chinese characteristics in the new era and the major issues to besolved at present.重点：The judgment method of singularity and pole; Using residue theoremKey Pointsto calculate residue扩充复平面上的留数定理；利用留数定理计算几种类型的实积分难点：The residue theorem on the complex plane is extended; Using residueFocal pointstheorem to calculate several types of real integrals知识单元序号支撑教学目标：61-6,2-1,2-4SLOs SupportedKnowledge Unit No.知识单元名称Fourier变换FouriertransformUnit TitleFourier积分Fourier integral知识点：Fourier变换FouriertransformKnowledge DeliveryFourier变换的性质PropertiesofFouriertransform卷积Convolution了解：Fourier变换的概念RecognizeThe concept of Fouriertransform理解：卷积和卷积定理UnderstandConvolution and convolution theorem学习目标：掌握Fourier变换的线性性质、位移性质、微分性质Learning Objectives积分性质掌握：Master the linear property,displacement property,Masterdifferential property and integral property of Fouriertransform2-1理解复变分析知识对于刻画工程实践问题的重要意义。Understand the significant meanings of the complex analysis indepictingthepractical engineeringproblems德育目标2-4注重科学思维方法的训练和科学伦理的教育，培养学生探索未Moral Objectives知、追求真理、勇攀科学高峰的责任感和使命感。Pay attention to the training of scientific thinking methods and theeducation of scientific ethics, and cultivatestudentssenseof10/14

10 / 14 The solution of residue at the pole; Residue theorem; A method of finding contour integral by residue 德育目标 Moral Objectives 2-2 让学生通过学习，掌握事物发展规律，通晓天下道理，丰富学 识，增长见识，塑造品格，努力成为德智体美劳全面发展的社会主 义建设者和接班人。 Let students master the law of development of things, understand the truth of the world, enrich their knowledge, increase their knowledge, shape their character, and strive to become socialist builders and successors with all-round development of morality, intelligence, physique, beauty and labor. 2-3 展示本专业在新时代中国特色社会主义建设中的成就和当前要 解决的重大课题。 Display the achievements of this major in the construction of socialism with Chinese characteristics in the new era and the major issues to be solved at present. 重点: Key Points The judgment method of singularity and pole; Using residue theorem to calculate residue 难点: Focal points 扩充复平面上的留数定理；利用留数定理计算几种类型的实积分 The residue theorem on the complex plane is extended; Using residue theorem to calculate several types of real integrals 知识单元序号: Knowledge Unit No. 6 支撑教学目标: SLOs Supported 1-6，2-1，2-4 知识单元名称 Unit Title Fourier 变换 Fourier transform 知识点: Knowledge Delivery Fourier 积分 Fourier integral Fourier 变换 Fourier transform Fourier 变换的性质 Properties of Fourier transform 卷积 Convolution 学习目标: Learning Objectives 了解: Recognize Fourier 变换的概念 The concept of Fourier transform 理解: Understand 卷积和卷积定理 Convolution and convolution theorem 掌握: Master 掌握 Fourier 变换的线性性质﹑位移性质﹑微分性质 ﹑积分性质 Master the linear property, displacement property, differential property and integral property of Fourier transform 德育目标 Moral Objectives 2-1 理解复变分析知识对于刻画工程实践问题的重要意义。 Understand the significant meanings of the complex analysis in depicting the practical engineering problems. 2-4 注重科学思维方法的训练和科学伦理的教育，培养学生探索未 知、追求真理、勇攀科学高峰的责任感和使命感。 Pay attention to the training of scientific thinking methods and the education of scientific ethics, and cultivate students' sense of