2016级复变函数教学大纲

FunctionsofaComplex总学时：64上机学时：0实验学时：04毕业要求（专业课程培养学生的教学内容与学时（一）复数及复平 4学 2学 2学程表示的几何图形掌握复数的乘幂与理解平面曲线,光滑曲线与平面区（二）复变函 8学 4学 4学C－RC－R及简单映射性质。掌握欧拉和复数的指数表示（三）复变函数的积 12学 8学柯西4学西积分定理和柯西积分。掌握柯西不等式、定理、莫勒拉定理。（四）级 16学 2学 6学 8学的求法。掌握泰勒级数的定义及常见函数的泰勒展开式。掌握洛朗级数的定（五）留 12学 4学 8学（六）保形映 12学 4学 4学 4学实验教学（包括上机学时、实验无现用：余家荣.复变函数[M].：高等教育方企勤.复变函数[M].：钟玉泉.复变函数论[M].：高等教育制定人及制定时间，2017220“FunctionsofaComplexVariable”CourseCourseFunctionsofaComplexCourseDisciplinaryBasicCourseCompulsoryClass4SchoolofProgramMathematicsandApplied Withasolidmathematicalfoundationandtheabilityofrigorouslogicalthinkingandreasoning.Masteringthebasictheoryofmathematicsandmethods,withtheabilitytousetheknowledgeofmathematicsandcomputerstosolveproblems.SubjecttopreliminarytraininginscientificByteachingthiscoursetoenablestudentstograspandunderstandsomebasicconcepts,basictheoriesandbasicmethodsofyticfunctiontheory,tocultivatethebasicskillsofapplyingtheseconceptsandmethodstosolvepracticalproblems,andtoprovidethenecessarymathematicalbasisforthestudyfollow-upCourseThiscourseisabasiccourseofmathematics,includingthebasictheory:complexnumbers,yticfunctions,differential,integral,series,residueandconformalmapofyticfunctions,andthebasicoperation:complexintegral,thepowerseriesexpansionofyticfunctions,classificationofapplicationoftheresiduetheorem,conformalmapofsimpleTeachingContentand (I)Complexnumberandcomplex 4ClassComplexnumberanditsgeometric 2ClassTopologyofthecomplexne 2ClassHoursMastertheconceptofcomplexnumbers,theirsrepresentationsandoperations.Understandthegeometricmeaningofarithmeticoperationsofcomplexnumbersandthegeometricalrepresentationofcomplexequation.Masterthepowerandtherootsofacomplexnumber.Understandthenecurve,smoothcurveneregion.Understandthedefinitionofcomplexfunctionofavariable,masterthelimitandcontinuityofcomplexFunctionsofacomplex 8Classytic 4ClassElementary 4ClassUnderstandingthederivabilityandthedifferentiabilityandclarifytherelationshipbetweenthetwoconcepts.MastertheC-Rconditionanduseittodeterminethedifferentiabilityofcomplexfunctions.Masterthebasicrulesofdifferentiation.Masterthedefinition,basicpropertiesandbasicmappropertiesoftheexponentialfunction,powerfunction,trigonometricfunction.MasterEulerformulaandexponentialrepresentationofcomplexnumbers.(III)Complex 12ClassCauchy's 8ClassCauchy'sintegralformula 4ClassHoursMasterthedefinition,basicpropertiesandcalculationmethodsoftheintegrationofacomplexfunctionalongapiecewisesmoothcurve.MasteringCauchyintegraltheoremandCauchyintegralformula.MasterCauchyinequality,Liouville'stheorem,Morera'stheorem.(IV) 16ClassThebasicpropertiesoftheseriesand 2ClassTheTaylor 6ClassTheLaurent 8ClassUnderstandtheconceptofthelimitofasequenceofcomplexnumbers,theconceptofseriesofcomplexfunctions.Mastertheconcept,thepropertiesandtheconvergenceradiusofthepowerseries.MasterthedefinitionoftheTaylorseriesandtheTaylorexpansionofcommonfunctions.MasterthedefinitionofLaurentseriesandtheLaurentexpansionofsomeyticfunctionsinanannulus.Masterthedefinitionandclassificationofisolatedsingularities.(V) 12ClassGeneral 4ClassApplicationofresiduecalculation 8ClassHoursMasterthedefinitionandcalculationmethodofresidues.Mastertheresiduetheoremanditsapplications.(VI)Conformal 12ClassHoursMappropertiesofunivalentytic 4ClassFractionallinearfunctionanditsmap 4ClassTheRiemannmap 4ClassUnderstandthegeometricmeaningofthederivativeofyticfunctionsandtheconceptofconformalmap;mastertheconceptoflinearfractionaltransformationanditsproperties;mastertheummodulusprinciple,Schwarzlemma,Riemannmaptheorem;mastertheconformalmapbetweensomesimpleClassroom ,comprehensiveExaminationandclassroomperformance:Finalexamination: andReferenceBooksTeachingMaterials:Ji