2016級(jí)復(fù)變函數(shù)教學(xué)大綱

FunctionsofaComplex總學(xué)時(shí)：64上機(jī)學(xué)時(shí)：0實(shí)驗(yàn)學(xué)時(shí)：04畢業(yè)要求（專業(yè)課程培養(yǎng)學(xué)生的教學(xué)內(nèi)容與學(xué)時(shí)（一）復(fù)數(shù)及復(fù)平 4學(xué) 2學(xué) 2學(xué)程表示的幾何圖形掌握復(fù)數(shù)的乘冪與理解平面曲線,光滑曲線與平面區(qū)（二）復(fù)變函 8學(xué) 4學(xué) 4學(xué)C－RC－R及簡(jiǎn)單映射性質(zhì)。掌握歐拉和復(fù)數(shù)的指數(shù)表示（三）復(fù)變函數(shù)的積 12學(xué) 8學(xué)柯西4學(xué)西積分定理和柯西積分。掌握柯西不等式、定理、莫勒拉定理。（四）級(jí) 16學(xué) 2學(xué) 6學(xué) 8學(xué)的求法。掌握泰勒級(jí)數(shù)的定義及常見(jiàn)函數(shù)的泰勒展開(kāi)式。掌握洛朗級(jí)數(shù)的定（五）留 12學(xué) 4學(xué) 8學(xué)（六）保形映 12學(xué) 4學(xué) 4學(xué) 4學(xué)實(shí)驗(yàn)教學(xué)（包括上機(jī)學(xué)時(shí)、實(shí)驗(yàn)無(wú)現(xiàn)用：余家榮.復(fù)變函數(shù)[M].：高等教育方企勤.復(fù)變函數(shù)[M].：鐘玉泉.復(fù)變函數(shù)論[M].：高等教育制定人及制定時(shí)間，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