2016年广播电视大学复变函数与积分变换重点公式归纳小抄

复变函数与积分变换复习提纲第一章复变函数一、复变数和复变函数YXIVUZFW,二、复变函数的极限与连续极限连续AFZLIM0LIM00ZFZ第二章解析函数一、复变函数可导与解析的概念。,YXIVUFW二、柯西黎曼方程掌握利用CR方程判别复变函数的可导性与解析性。XYV掌握复变函数的导数YXYXVIIUFVFZF1三、初等函数重点掌握初等函数的计算和复数方程的求解。1、幂函数与根式函数单值函数INNNERIRIRZWSCOSCOK0、1、2、N1N多值函数NKZIE2AG12、指数函数SINCYXZ性质（1）单值（2）复平面上处处解析，（3）以为周期ZEI23、对数函数（K0、1、2）KIZKZILNZW2LN2ARGL性质（1）多值函数,（2）除原点及负实轴处外解析,（3）在单值解析分枝上。KZ1LN4、三角函数2COSIZIEZIEIZI2SIN性质（1）单值（2）复平面上处处解析（3）周期性（4）无界5、反三角函数（了解）反正弦函数1SIN2ZILZARCW反余弦函数O2I性质与对数函数的性质相同。6、一般幂函数（K0、1）ARG2LNIZKZSSLZE四、调和函数与共轭调和函数1调和函数0,2YXU2已知解析函数的实部（虚部），求其虚部（实部）有三种方法A）全微分法B）利用CR方程C）不定积分法第三章解析函数的积分一、复变函数的积分存在的条件。LLLUDYVXIDYUXFDZ二、复变函数积分的计算方法1、沿路径积分利用参数法积分，关键是写出路径的参数方程。CF2、闭路积分A利用留数定理，柯西积分公式，高阶导数公式。DZB利用参数积分方法ZYXIVUC,三、柯西积分定理0CDZF推论1积分与路径无关DZFFZC21推论2利用原函数计算积分1221FZFZ推论3二连通区域上的柯西定理21CCDZFF推论4复连通区域上的柯西定理KCNCZFZF1四、柯西积分公式DZFIZFC2002CFZDIFZA五、高阶导数公式FINFCNN1解析函数的两个重要性质解析函数在任一点的值可以通过函数沿包围点的任一简单闭合回路的积分表示。ZFZZ解析函数有任意阶导数。本章重点掌握复变函数积分的计算方法沿路径积分1）利用参数法积分2）利用原函数计算积分。CDZF闭路积分利用留数定理计算积分。第四章解析函数的级数一、幂级数及收敛半径0NNBZA1、一个收敛半径为R（0）的幂级数，在收敛圆内的和函数是解析函数，在这个收敛圆内，这个展开式ZF可以逐项积分和逐项求导，即有1NNBZAZFRBZ1000NNZLNZADDFRBZ2、收敛半径的计算方法1）比值法1/LIMNNR2）根值法A二、泰勒（TAYLOR）级数1、如函数在圆域内解析，那么在此圆域内可以展开成TAYLOR级数ZFRBZFFNNNBZFA001）展开式是唯一的。故将函数在解析点的邻域中展开幂级数一定是TAYLOR级数。2收敛半径是展开点到的所有奇点的最短距离。ZF3）展开式的系数可以微分计算NBFA4）解析函数可以用TAYLOR级数表示。2、记住一些重要的泰勒级数1）2）0NZ0NZE3）4）012SINN021COSNNZZ三、罗兰（LAURENT）级数如果函数在圆环城内解析，则ZF21RBZZFXNNBCDZBFICLNN12（N0、1、2）1、展开式是唯一的，即只要把函数在圆环城内展开为幂级数即为LAURENT级数。2、展开式的系数是不可以利用积分计算。利用已知的幂级数，通过代数运算把函数展开成LAURENT级数。3、注意展开的区域，在展开点的所有解析区域展开。四、孤立奇点1、定义若B是的孤立奇点，则在内解析。在此点可展开为罗兰级数，ZFZFB0ZFZF10NNNNCBCZC2、分类孤立奇点1,RE,CBZFSF无穷多负幂项本性奇点有限负幂项极点无负幂项可去奇点把函数在奇点的去心邻域中展开为罗兰级数，求解C13、极点留数计算A如果B是的一阶极点，则ZFLIM,EZFZFSBZB如果B是的M阶极点，则LI1,RE1ZFBDZZFSMBC如B是的一阶极点，且PB0，那么ZQPFBS,RED0,1E,2ZFSZFE若是的可去奇点，并且，0LIMZFZCZFSZLI,RE1关系全平面留数之和为零。0,RE,1FSBZFSKK本章重点函数展开成TAYLOR级数，并能写出收敛半径。函数在解析圆环城内展开成LAURENT级数。孤立奇点（包含点）的判定及其留数的计算。Z第五章留数定理的应用一、DR20COS,IN条件（1）RSIN，COS为COS与SIN的有理函数（2）R在0，2或者，上连续。令，则，。IEZIZSIN12COS1ZIZDZFIIZRDZZ1120,CO,INKKFSI1,E2K注意留数是计算单位圆中的奇点。二、DXF条件（1）是X的多项式。XQP,（2）0X（3）分母阶次比分子阶次至少高二次则是在上半平面的奇点。NRKBZFSIDF1,EZF三、（）XEI0条件（1），且比至少高一阶，XQPXP（2），（3）XNKKZIIBERSIDERI1,0IMK，IXCOSXDSIN重点关注第一和第三种类型第七章FOURIER变换一、傅立叶变换DTEXFFJFTJ21二、函数的傅立叶变换1DXEXJXDEXJ2三、一些傅立叶变换及逆变换IXH121四、性质FXF1、相似性质AF12、延迟性质FEXXJ00位移性质0FEJ3、微分性质FJXFFXJFJFNNNNDFJ4、积分性质10FJDXFX由FOURIER变换的微分和积分性质，我们可以利用FOURIER变换求解微积分方程。四、卷积和卷积定理DXFXF2121F2211XF五、三维FOURIER变换及反演本章重点利用定义计算FOURIER变换第八章LAPLACE变换一、拉普拉斯变换PFDTEXFFP0二、几个重要的拉普拉斯变换及逆变换PTH1TH1ETTEP12COSPTTCOS212SINPTTPSIN211MTT四、拉普拉斯变换的性质1、PFETFT002、0TP3、FTF2FPFUNUNDTF4、PFTFT10DTF五、卷积TFTFT20121PF六、LAPLACE反演NNPTPTJESDETF1,R21七、LAPLACE逆变换（1）部分分式法（2）卷积定理（3）LAPLACE反演公式（留数定理）（4）利用LAPLACE变换的性质八、利用LAPLACE变换求解微积分方程（1）对方程取LAPLACE变换，得到象函数的代数方程（2）解代数方程，得到像函数的表达式（3）求像函数的拉普拉斯逆变换拉氏变换解代数方程拉氏逆变换本章重点利用定义和性质计算LAPLACE变换。微分方程像函数的代数方程像函数像原函数解函数计算LAPLACE逆变换。利用LAPLACE变换求解微积分方程。请您删除一下内容，O_O谢谢2016年中央电大期末复习考试小抄大全，电大期末考试必备小抄，电大考试必过小抄BASKETBALLCANMAKEATRUECLAIMTOBEINGTHEONLYMAJORSPORTTHATISANAMERICANINVENTIONFROMHIGHSCHOOLTOTHEPROFESSIONALLEVEL,BASKETBALLATTRACTSALARGEFOLLOWINGFORLIVEGAMESASWELLASTELEVISIONCOVERAGEOFEVENTSLIKETHENATIONALCOLLEGIATEATHLETICASSOCIATIONNCAAANNUALTOURNAMENTANDTHENATIONALBASKETBALLASSOCIATIONNBAANDWOMENSNATIONALBASKETBALLASSOCIATIONWNBAPLAYOFFSANDITHASALSOMADEAMERICANHEROESOUTOFITSPLAYERANDCOACHLEGENDSLIKEMICHAELJORDAN,LARRYBIRD,EARVIN“MAGIC“JOHNSON,SHERYLSWOOPES,ANDOTHERGREATPLAYERSATTHEHEARTOFTHEGAMEISTHEPLAYINGSPACEANDTHEEQUIPMENTTHESPACEISARECTANGULAR,INDOORCOURTTHEPRINCIPALPIECESOFEQUIPMENTARETHETWOELEVATEDBASKETS,ONEATEACHENDINTHELONGDIRECTIONOFTHECOURT,ANDTHEBASKETBALLITSELFTHEBALLISSPHERICALINSHAPEANDISINFLATEDBASKETBALLSRANGEINSIZEFROM28530IN7276CMINCIRCUMFERENCE,ANDINWEIGHTFROM1822OZ510624GFORPLAYERSBELOWTHEHIGHSCHOOLLEVEL,ASMALLERBALLISUSED,BUTTHEBALLINMENSGAMESMEASURES29530IN7576CMINCIRCUMFERENCE,ANDAWOMENSBALLIS28529IN7274CMINCIRCUMFERENCETHECOVERINGOFTHEBALLISLEATHER,RUBBER,COMPOSITION,ORSYNTHETIC,ALTHOUGHLEATHERCOVERSONLYAREDICTATEDBYRULESFORCOLLEGEPLAY,UNLESSTHETEAMSAGREEOTHERWISEORANGEISTHEREGULATIONCOLORATALLLEVELSOFPLAY,THEHOMETEAMPROVIDESTHEBALLINFLATIONOFTHEBALLISBASEDONTHEHEIGHTOFTHEBALLSBOUNCEINSIDETHECOVERINGORCASING,ARUBBERBLADDERHOLDSAIRTHEBALLMUSTBEINFLATEDTOAPRESSURESUFFICIENTTOMAKEITREBOUNDTOAHEIGHTMEASUREDTOTHETOPOFTHEBALLOF4954IN1214MWHENITISDROPPEDONASOLIDWOODENFLOORFROMASTARTINGHEIGHTOF6FT180MMEASUREDFROMTHEBOTTOMOFTHEBALLTHEFACTORYMUSTTESTTHEBALLS,ANDTHEAIRPRESSURETHATMAKESTHEBALLLEGALINKEEPINGWITHTHEBOUNCETESTISSTAMPEDONTHEBALLDURINGTHEINTENSITYOFHIGHSCHOOLANDCOLLEGETOURNEYSANDTHEPROFESSIONALPLAYOFFS,THISINFLATEDSPHERECOMMANDSCONSIDERABLEATTENTIONBASKETBALLISONEOFFEWSPORTSWITHAKNOWNDATEOFBIRTHONDECEMBER1,1891,INSPRINGFIELD,MASSACHUSETTS,JAMESNAISMITHHUNGTWOHALFBUSHELPEACHBASKETSATTHEOPPOSITEENDSOFAGYMNASIUMANDOUTLINED13RULESBASEDONFIVEPRINCIPLESTOHISSTUDENTSATTHEINTERNATIONALTRAININGSCHOOLOFTHEYOUNGMENSCHRISTIANASSOCIATIONYMCA,WHICHLATERBECAMESPRINGFIELDCOLLEGENAISMITH18611939WASAPHYSICALEDUCATIONTEACHERWHOWASSEEKINGATEAMSPORTWITHLIMITEDPHYSICALCONTACTBUTALOTOFRUNNING,JUMPING,SHOOTING,ANDTHEHANDE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