复变函数论-第七章-共形映射

为搞好山东省交通科学研究所研发基地项目的结算审计工作,我跟踪审计部特针对本项目作如下要求,请各施工单位、供货单位遵照执行：andperformancetestcopiesoftherecord.Ifnecessary,reviewshouldbecarriedout;4)forspringhangers(includedsimplespring,hangersandconstantsupporthangers)itshouldalsoberecognizedassettingandlockingofloads.5)checkthesurfacequality,foldedlayeringandwithoutcracks,rustandotherdefects.5)aftercompletionofthetestandcontroldrawingnumberonebyone,byseriesbaled.Coloralloysteelparts,thepartsmarkinginstallationlocationandrotationaboutthedirectionyouwant.7.3.14.hangersinstallation7.3.14.1hangerlayouta.acleardesignofhangershouldbeinstalledstrictlyinaccordancewiththedrawingsanddesignsshallnotbeinstalledwrong,missing,etc.B.ownarrangementofpipingsupportandhangersetandselectionshouldbebasedoncomprehensiveanalysisofgenerallayoutofpipingsystems;coldinstallationofsteampipewithparticularattentionreservedforcompensationofthermalexpansiondisplacementandorientation.C.supportsystemsshouldberationaltowithstandpipeloads,staticloadandincidentalload;reasonablepipingdisplacement;guaranteedundervariousconditions,stressarewithintheallowedrange.Strength,stiffness,andmeetrequirementstopreventvibrationandsoothingwater,withoutaffectingtheadjacentequipmentmaintenanceandotherpipinginstallationandexpansion.D.equipmentconnectedtotheinterfacetomeetpipelinethrust(torque)limitrequirements;increasethestabilityofpipingsystemstopreventpipeline...Tubewallthickness(mm)2-34-67-10weldformnoslopemouthweldstrengtheningheighth(mm)1-1.51.5-2weldwidthb(mm)5-67-6hasslopemouthweldstrengtheningheighth(mm)1.5-22weldwidthb(mm)coverhadeachedgeslopemouthabout2mmargonarcweldingweldstrengtheningsurfaceheightandwidthtubewallthickness(mm)2--33--45--6weldformweldstrengtheningheighth(mm)1-1.51.5-22-2.5widthb(mm)7．1解析函数的特性教学目的：使学生掌握从映射角度来研究解析函数的概念及基本原理，从而了解解析函数的几何理论.重点：保角映射的概念与性质.难点：解析变换的保域性.课时：4课时教学过程：前几章我们用分析的方法研究了解析函数的性质和应用，从映射角度来研究解析函数的性质及其应用主是通常说的解析函数的几何理论.几何理论中最基本的是共形映射的理论.下面我们来介绍共形映射的概念及基本原理.一．解析函数的保域性.定理7.1(保域定理)设()wfz在区域D内解析且不恒为常数,则D的象()GfD也是一个区域.证明：按区域的定义：要证()GfD是一个连通开集.首先证明G是一个开集即证G的每一个都是内点,设0w是G内的任意一点，则存在0zD，使得00()fzw，由第六章的儒歇定理，必存在0w的一个邻域*0ww.对于其中的任一数wA，函数()fzA在0zz内（0zz是D内的邻域）必有根，即wA，这记0wwG.表明0w是G的内点.由0w的任意性知G是开集其次证明G是连通集.由于D是区域,可在D内部取一条联结12,zz的折线121122:()[,(),()]Czzttttztzztz.于是：12:[()]()wfztttt就是联结12,ww的并且完全含于G的一条曲线.从而,由柯西积分定理的古莎证明第三步,可以找到一条联结12,ww内接于且完全含于G的折线.从以上两点，表明()GfD是区域.推论7.2设()wfz在区域D内单叶解析,则D的象()GfD也是一个区域.证明：用()fz在区域D内单叶，必()fz在D内不恒为常数.定理7.3设函数()wfz在点0z解析,且'0()0fz,则()fz在0z的一个邻域内单叶解析.由此可见，符合本定理条件的解析变换()wfz将0z的一个充分小邻域变成00()wfz的为搞好山东省交通科学研究所研发基地项目的结算审计工作,我跟踪审计部特针对本项目作如下要求,请各施工单位、供货单位遵照执行：andperformancetestcopiesoftherecord.Ifnecessary,reviewshouldbecarriedout;4)forspringhangers(includedsimplespring,hangersandconstantsupporthangers)itshouldalsoberecognizedassettingandlockingofloads.5)checkthesurfacequality,foldedlayeringandwithoutcracks,rustandotherdefects.5)aftercompletionofthetestandcontroldrawingnumberonebyone,byseriesbaled.Coloralloysteelparts,thepartsmarkinginstallationlocationandrotationaboutthedirectionyouwant.7.3.14.hangersinstallation7.3.14.1hangerlayouta.acleardesignofhangershouldbeinstalledstrictlyinaccordancewiththedrawingsanddesignsshallnotbeinstalledwrong,missing,etc.B.ownarrangementofpipingsupportandhangersetandselectionshouldbebasedoncomprehensiveanalysisofgenerallayoutofpipingsystems;coldinstallationofsteampipewithparticularattentionreservedforcompensationofthermalexpansiondisplacementandorientation.C.supportsystemsshouldberationaltowithstandpipeloads,staticloadandincidentalload;reasonablepipingdisplacement;guaranteedundervariousconditions,stressarewithintheallowedrange.Strength,stiffness,andmeetrequirementstopreventvibrationandsoothingwater,withoutaffectingtheadjacentequipmentmaintenanceandotherpipinginstallationandexpansion.D.equipmentconnectedtotheinterfacetomeetpipelinethrust(torque)limitrequirements;increasethestabilityofpipingsystemstopreventpipeline...Tubewallthickness(mm)2-34-67-10weldformnoslopemouthweldstrengtheningheighth(mm)1-1.51.5-2weldwidthb(mm)5-67-6hasslopemouthweldstrengtheningheighth(mm)1.5-22weldwidthb(mm)coverhadeachedgeslopemouthabout2mmargonarcweldingweldstrengtheningsurfaceheightandwidthtubewallthickness(mm)2--33--45--6weldformweldstrengtheningheighth(mm)1-1.51.5-22-2.5widthb(mm)一个曲边邻域.2解析变换的保角——导数的几何意义设()wfz于区域D内解析,0zD,在点0z有导数0z.通过0z任意引一条有向光滑曲线01:()()Czztttt,00()zzt，则必0'()zt存在且0'()0zt，从而由第二章习题（一）1，C在0z有切线，0'()zt就是切向量，它的倾角为0arg'()zt.经过变换()wfz，C之象曲线()fC的参数方程应为01:[()]()wfztttt由定理7.3及第三章习题（一）13，在点0wt0w＝（）的邻域内是光滑的，又由于000'()'()'()0wtfzzt，故在00()wfz也有切线，0'()wt就是切向量，其倾角为000arg'()arg'()arg'(),wtfzzt即0arg'()fz假设0'()Riafze则必00'(),arg'()fzRfza，于是a（7.1）且lim0zwRz（7.2）图7.1假定x轴与u轴、y轴与v轴的正方向相同（如图7.1），而且将原曲线的切线正方向与变换后象曲线的切线正方向间的夹角，理解为原曲线经过变换后的旋转角，则（7.1）说明:象曲线在点00()wfz的切线正向,可由原象曲线C在点0z的切线为搞好山东省交通科学研究所研发基地项目的结算审计工作,我跟踪审计部特针对本项目作如下要求,请各施工单位、供货单位遵照执行：andperformancetestcopiesoftherecord.Ifnecessary,reviewshouldbecarriedout;4)forspringhangers(includedsimplespring,hangersandconstantsupporthangers)itshouldalsober