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《复变函数与积分变换》作业参考答案习题
14、计算下列各式1;3;5,求,,;7解1;3;5,,.7因为,所以,即时,;时,;时,;时,;时,;时,.习题
23、下列函数在何处可导?何处解析?在可导点求出其导数.2;46解2因为,,,,,.这四个一阶偏导数都连续,故和处处可微,但柯西-黎曼方程仅在上成立,所以只在直线上可导,此时,但复平面上处处不解析.4因为,,,,,.这四个一阶偏导数都连续,故和处处可微,且满足柯西-黎曼方程,所以在复平面内解析,并且.6所以,在除外处处解析,且.
4、指出下列函数的奇点.1;2.解1所以,的奇点为0,.2所以,的奇点为,.
10、如果在区域内解析,并且满足下列条件之一,试证在内是一常数.2在内解析;证明由在区域内解析,知、在区域内可微,且,.同理,由在内解析,知,.从而我们得到,所以、皆为常数,故在内是一常数.
15、求解下列方程2解,于是
18、求,的值及主值.解,所以其主值为;,所以其主值为.
19、求,,,的值.解;;;.
20、求,,,,的值.解;;;;
22、解方程1;解,.习题
31、沿下列路径计算积分1从原点至的直线段;2从原点沿实轴至2,再由2铅直向上至;3从原点沿虚轴至,再由沿水平方向向右至.解1从原点至的直线段的复参数方程为,,参数,所以2从原点沿实轴至2的直线段的复参数方程为,参数,由2铅直向上至的直线段的复参数方程为,参数,所以3从原点沿虚轴至的直线段的复参数方程为,参数,由沿水平方向向右至的复参数方程为,参数,所以
2、分别沿与算出积分的值.解的复参数方程为,,参数所以;的复参数方程为,,参数所以
5、计算积分的值,其中为正向圆周1解设是内以被积函数的奇点为圆心的正向圆周,那么
6、试用观察法得出下列积分的值,并说明观察时所依据的是什么?是正向圆周1;2;3;4;5;6.解1,根据柯西积分定理;2,根据柯西积分定理;3,根据柯西积分定理;4,根据复合闭路定理;5,根据柯西积分定理;6,根据柯西积分定理及复合闭路定理.
7、沿指定曲线的正向计算下列积分1,;2,;3,;4,;5,;6,为包围的闭曲线;7,;8,;9,;10,.解1;2;3;4;5;6;7;8;9;10.
21、证明和都是调和函数,但是不是解析函数.证明因为,,,,,,,,所以,,且,.即和都是调和函数,但是不是解析函数.
22、由下列各已知调和函数求解析函数,并写出的表达式1;2,;3,.解1因为是调和函数,所以,.于是.那么,则,所以,2,.因为是调和函数,所以,从而.由知,所以.3因为是调和函数,所以,.于是.那么,则,所以,由知,所以.习题
41、下列数列是否收敛?若收敛,求其极限.1;2;3;4.解1,当时,实部,虚部,所以收敛于.2,当时,那么,所以收敛于0.3当时,实部是发散的,所以发散.4,实部和虚部都发散,所以发散.
2、判断下列级数的收敛性与绝对收敛性1;3.解1记,则当时,那么不趋近于0,所以级数发散.3收敛,即级数绝对收敛,所以收敛.
7、将下列各函数展成的幂级数,并指出它们的收敛半径.1;3.解1.因为,所以收敛半径.3因为,所以收敛半径.
8、将下列各函数在指定点处展成泰勒级数,并指出它们的收敛半径.3,;4,;6,.解3,则.因为,所以收敛半径.4,则.因为,所以收敛半径.6.因为,所以收敛半径.
10、求下列各函数在指定圆环域的洛朗级数展开式2,,;5,在以为中心的圆环域内;7,.解2在内,由于,且,所以,从而.在内,由于,所以,从而.5当时,由于,且,所以,从而.当时,由于,所以,且,从而,所以.7由于且,所以习题
51、求下列函数的孤立奇点并确定它们的类别,若是极点,指出它们的级.1;3;4;7;11.解1易见,是的孤立奇点.由于,,所以,是极点.,一级极点,,二级极点.3,所以是极点.,二级极点.4易见是的孤立奇点,且,所以是可去奇点;7,三级极点,,一级极点;11,本性奇点.
5、求下列各函数在有限奇点处的留数.2;3;6.解2记,则易见,是的孤立奇点,且他们都是一级极点.由规则Ⅰ,,,.3记,则有二级极点.由规则Ⅱ,,.6记,则有本性奇点.因为在的去心邻域内的洛朗级数为于是有其中的项的系数,所以
6、利用留数定理计算下列积分.1,为圆周解被积函数在圆周的内部有一级极点和二级极点,由留数的计算规则Ⅰ、Ⅱ得,.于是由留数定理得积分值2解被积函数在内有一个二级极点,由留数的计算规则Ⅱ得于是由留数定理得积分值4解被积函数在内有可去奇点,则,所以由留数定理知6解被积函数在内有一个二级极点,由留数的计算规则Ⅱ得于是由留数定理得积分值
9、1解令,则,.于是被积函数在内有一个一级极点,其留数所以5解是偶函数,而在上半平面内有一级极点和,且,,所以6解,,,,且在实轴上无孤立奇点,故积分存在,所求积分是它的实部.函数在上半平面有两个一级极点和,而且,,从而所以习题
84、试求的傅氏变换.解的傅里叶变化为
5、试求矩形脉冲的傅氏变换.解的傅里叶变化为
6、求下列函数的傅氏积分1解是上的奇函数,则,,于是
7、求函数的傅氏积分,并计算.解是上的偶函数,则,,于是
10、求符号函数的傅氏变换.(提示.)解方法一.方法二.
11、求函数的傅氏变换.解,则
15、利用位移性质计算下列函数的傅氏变换1;2解1;2.
23、求下列函数的傅氏变换2;3;4.解2记,,由卷积定理有3记,,由卷积定理有4记,,由卷积定理有习题
92、求下列函数的拉氏变换
13.解1.3.
3、求下列周期函数的拉氏变换1以为周期且在一个周期内的表达式为.解
4、求下列函数的拉氏变换1;2;3;6(为实常数);9;10;11.解123;6,则由位移性质有;9,则;10,则,从而;11,则.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BasketballcanmakeatrueclaimtobeingtheonlymajorsportthatisanAmericaninvention.FromhighschooltotheprofessionallevelbasketballattractsalargefollowingforlivegamesaswellastelevisioncoverageofeventsliketheNationalCollegiateAthleticAssociationNCAAannualtournamentandtheNationalBasketballAssociationNBAandWomensNationalBasketballAssociationWNBAplayoffs.AndithasalsomadeAmericanheroesoutofitsplayerandcoachlegendslikeMichaelJordanLarryBirdEarvinMagicJohnsonSherylSwoopesandothergreatplayers.Attheheartofthegameistheplayingspaceandtheequipment.Thespaceisarectangularindoorcourt.Theprincipalpiecesofequipmentarethetwoelevatedbasketsoneateachendinthelongdirectionofthecourtandthebasketballitself.Theballissphericalinshapeandisinflated.Basket-ballsrangeinsizefrom
28.5-30in72-76cmincircumferenceandinweightfrom18-22oz510-624g.Forplayersbelowthehighschoollevelasmallerballisusedbuttheballinmensgamesmeasures
29.5-30in75-76cmincircumferenceandawomensballis
28.5-29in72-74cmincircumference.Thecoveringoftheballisleatherrubbercompositionorsyntheticalthoughleathercoversonlyaredictatedbyrulesforcollegeplayunlesstheteamsagreeotherwise.Orangeistheregulationcolor.Atalllevelsofplaythehometeamprovidestheball.Inflationoftheballisbasedontheheightoftheballsbounce.Insidethecoveringorcasingarubberbladderholdsair.Theballmustbeinflatedtoapressuresufficienttomakeitreboundtoaheightmeasuredtothetopoftheballof49-54in
1.2-
1.4mwhenitisdroppedonasolidwoodenfloorfromastartingheightof6ft
1.80mmeasuredfromthebottomoftheball.Thefactorymusttesttheballsandtheairpressurethatmakestheballlegalinkeepingwiththebouncetestisstampedontheball.Duringtheintensityofhighschoolandcollegetourneysandtheprofessionalplayoffsthisinflatedspherecommandsconsiderableattention.Basketballisoneoffewsportswithaknowndateofbirth.OnDecember11891inSpringfieldMassachusettsJamesNaismithhungtwohalf-bushelpeachbasketsattheoppositeendsofagymnasiumandout-lined13rulesbasedonfiveprinciplestohisstudentsattheInternationalTrainingSchooloftheYoungMensChristianAssociationYMCAwhichlaterbecameSpringfieldCollege.Naismith1861-1939wasaphysicaleducationteacherwhowasseekingateamsportwithlimitedphysicalcontactbutalotofrunningjumpingshootingandthehand-eyecoordinationrequiredinhandlingaball.Thepeachbasketshehungasgoalsgavethesportthenameofbasketball.HisstudentswereexcitedaboutthegameandChristmasvacationgavethemthechancetotelltheirfriendsandpeopleattheirlocalYMCAsaboutthegame.TheassociationleaderswrotetoNaismithaskingforcopiesoftherulesandtheywerepublishedintheTriangletheschoolnewspaperonJanuary
151892.Naismithsfivebasicprinciplescenterontheballwhichwasdescribedaslargelightandhandledwiththehands.Playerscouldnotmovetheballbyrunningaloneandnoneoftheplayerswasrestrictedagainsthandlingtheball.Theplayingareawasalsoopentoallplayersbuttherewastobenophysicalcontactbetweenplayers;theballwastheobjective.Toscoretheballhadtobeshotthroughahorizontalelevatedgoal.Theteamwiththemostpointsattheendofanallottedtimeperiodwins.EarlyinthehistoryofbasketballthelocalYMCAsprovidedthegymnasiumsandmembershipintheorganizationgrewrapidly.Thesizeofthelocalgymdictatedthenumberofplayers;smallergymsusedfiveplayersonasideandthelargergymsallowedseventonine.Theteamsizebecamegenerallyestablishedasfivein1895andin1897thiswasmadeformalintherules.TheYMCAlostinterestinsupportingthegamebecause10-20basketballplayersmonopolizedagymnasiumpreviouslyusedbymanymoreinavarietyofactivities.YMCAmembershipdroppedandbasketballenthusiastsplayedinlocalhalls.Thisledtothebuildingofbasketballgymnasiumsatschoolsandcollegesandalsototheformationofprofessionalleagues.AlthoughbasketballwasbornintheUnitedStatesfiveofNaismithsoriginalplayerswereCanadiansandthegamespreadtoCanadaimmediately.ItwasplayedinFranceby1893;Englandin1894;AustraliaChinaandIndiabetween1895and1900;andJapanin
1900.From1891through1893asoccerballwasusedtoplaybasketball.Thefirstbasketballwasmanufacturedin
1894.Itwas32in81cmincircumferenceorabout4in10cmlargerthanasoccerball.Thededicatedbasketballwasmadeoflacedleatherandweighedlessthan20oz567g.Thefirstmoldedballthateliminatedtheneedforlaceswasintroducedin1948;itsconstructionandsizeof30in76cmwereruledofficialin
1949.Therule-setterscamefromseveralgroupsearlyinthe1900s.Collegesanduniversitiesestablishedtheirrulescommitteesin1905theYMCAandtheAmateurAthleticUnionAAUcreatedasetofrulesjointlystatemilitiagroupsabidedbyasharedsetofrulesandthereweretwoprofessionalsetsofrules.AJointRulesCommitteeforcollegestheAAUandtheYMCAwascreatedin1915andunderthenametheNationalBasketballCommitteeNBCmaderulesforamateurplayuntil
1979.InthatyeartheNationalFederationofStateHighSchoolAssociationsbegangoverningthesportatthehighschoollevelandtheNCAARulesCommitteeassumedrule-makingresponsibilitiesforjuniorcollegescollegesandtheArmedForceswithasimilarcommitteeholdingjurisdictionoverwomensbasketball.UntilWorldWarIIbasketballbecameincreasinglypopularintheUnitedStatesespeciallyatthehighschoolandcollegelevels.AfterWorldWarIIitspopularitygrewaroundtheworld.Inthe1980sinterestinthegametrulyexplodedbecauseoftelevisionexposure.BroadcastoftheNCAAChampionshipGamesbeganin1963andbythe1980scabletelevisionwascarryingregularseasoncollegegamesandevenhighschoolchampionshipsinsomestates.PlayerslikeBillRussellWiltChamberlainandLewAlcindorKareemAbdul-Jabbarbecamenationallyfamousatthecollegelevelandcarriedtheirfansalongintheirprofessionalbasketballcareers.Thewomensgamechangedradicallyin1971whenseparaterulesforwomenweremodifiedtomorecloselyresemblethemensgame.TelevisioninterestfollowedthewomenaswellwithbroadcastofNCAAchampionshiptourneysbeginningintheearly1980sandtheformationoftheWNBAin
1997.InternationallyItalyhasprobablybecometheleadingbasketballnationoutsideoftheUnitedStateswithnationalcorporateandprofessionalteams.TheOlympicsboostsbasketballinternationallyandhasalsospurredthewomensgamebyrecognizingitasanOlympiceventin
1976.AgaintelevisioncoverageoftheOlympicshasbeenexceptionallyimportantindrawingattentiontointernationalteams.ThefirstprofessionalmensbasketballleagueintheUnitedStateswastheNationalBasketballLeagueNBLwhichdebutedin
1898.Playerswerepaidonaper-gamebasisandthisleagueandotherswerehurtbythepoorqualityofgamesandtheever-changingplayersonateam.AftertheGreatDepressionanewNBLwasorganizedin1937andtheBasketballAssociationofAmericawasorganizedin
1946.Thetwoleaguescametoagreethatplayershadtobeassignedtoteamsonacontractbasisandthathighstandardshadtogovernthegame;underthesepremisesthetwojoinedtoformtheNationalBasketballAssociationNBAin
1949.ArivalAmericanBasketballAssociationABAwasinauguratedin1967andchallengedtheNBAforcollegetalentandmarketshareforalmosttenyears.In1976thisleaguedisbandedbutfourofitsteamsremainedasNBAteams.Unificationcamejustintimeformajortelevisionsupport.SeveralwomensprofessionalleagueswereattemptedandfailedincludingtheWomensProfessionalBasketballLeagueWBLandtheWomensWorldBasketballAssociationbeforetheWNBAdebutedin1997withthesupportoftheNBA.JamesNaismithoriginallyfromAl-monteOntarioinventedbasketballattheInternationalYMCATrainingSchoolinSpringfieldMassachusettsin
1891.Thegamewasfirstplayedwithpeachbasketshencethenameandasoccerballandwasintendedtoprovideindoorexerciseforfootballplayers.Asaresultitwasoriginallyaroughsport.AlthoughtenofNaismithsoriginalthirteenrulesremainthegamesoonchangedconsiderablyandthefounderhadlittletodowithitsevolution.ThefirstintercollegiategamewasplayedinMinnesotain1895withnineplayerstoasideandafinalscoreofninetothree.Ayearlaterthefirstfive-manteamsplayedattheUniversityofChicago.Basketswerenowconstructedoftwinenetsbutitwasnotuntil1906thatthebottomofthenetswereopen.In1897thedribblewasfirstusedfieldgoalsbecametwopointsfoulshotsonepointandthefirstprofessionalgamewasplayed.AyearlaterthefirstprofessionalleaguewasstartedintheEastwhilein1900thefirstintercollegiateleaguebegan.In1910inordertolimitroughplayitwasagreedthatfourfoulswoulddisqualifyplayersandglassbackboardswereusedforthefirsttime.NonethelessmanyrulesstilldiffereddependinguponwherethegameswereplayedandwhetherprofessionalscollegiansorYMCAplayerswereinvolved.CollegebasketballwasplayedfromTexastoWisconsinandthroughouttheEastthroughthe1920sbutmostteamsplayedonlyintheirownregionswhichpreventedanationalgameoraudiencefromdeveloping.ProfessionalbasketballwasplayedalmostexclusivelyintheEastbeforethe1920sexceptwhenateamwouldbarnstormintotheMidwesttoplaylocalteamsoftenafteraleaguehadfolded.Beforethe1930sveryfewgameseitherprofessionaloramateurwereplayedinfacilitiessuitableforbasketballorwithaperfectlyroundball.Somewereplayedinarenaswithchickenwireseparatingtheplayersfromfansthusthewordcagersotherswithpostsinthemiddleofthefloorandoftenwithbalconiesoverhangingthecornerslimitingtheareasfromwhichshotscouldbetaken.Untilthelate1930sallplayersusedthetwo-handsetshotandscoresremainedlow.Basketballinthe1920sand1930sbecamebothmoreorganizedandmorepopularalthoughitstilllaggedfarbehindbothbaseballandcollegefootball.Intheprosfiveurbanethnicteamsexcelledandplayedwithalmostnocollegegraduates.TheyweretheNewYorkOriginalCeltics;theClevelandRosenblumsownedbyMaxRosenblum;EddieGottliebsPhiladelphiaSPHAsSouthPhiladelphiaHebrewAssociation;andtwogreatblackteamstheNewYorkRenaissanceFiveandAbeSapersteinsHarlemGlobetrotterswhichwasactuallyfromChicago.WhiletheseteamshadsomenotableplayersnosuperstarssuchasBabeRuthJackDempseyorRedGrangeemergedtocapturethepublicsattentionastheydidinothersportsoftheperiod.Thesamewastrueincollegebasketballupuntilthelate1930swithcoachesdominatingthegameanditsdevelopment.WalterDocMeanwellatWisconsinForrestPhogAllenatKansasWardPiggyLambertatPurdueandHenryDocCarlsonatPittsburghallmadesignificantcontributionstothegamesdevelopment:zonedefensestheweavethepassinggameandthefastbreak.InthedecadeprecedingWorldWarIIfiveeventschangedcollegebasketballandallowedittobecomeamajorspectatorsport.In1929therulescommitteereversedadecisionthatwouldhaveoutlaweddribblingandslowedthegameconsiderably.FiveyearslaterpromoterEdwardNedIrishstagedthefirstintersectionaltwinbillinMadisonSquareGardeninNewYorkCityandattractedmorethan16000fans.HedemonstratedtheappealofmajorcollegeballandmadeNewYorkitscenter.InDecember1936HankLuisettiofStanfordrevealedthevirtuesoftheone-handedshottoanamazedGardenaudienceandbecamethefirstmajorcollegiatestar.SoonthereafterLuisettiscoredanincrediblefiftypointsagainstDuquesnethusendingtheEastsdevotiontothesetshotandencouragingamoreopengame.Inconsecutiveyearsthecenterjumpwaseliminatedafterfreethrowsandthenafterfieldgoalsthusspeedingupthegameandallowingformorescoring.In1938IrishcreatedtheNationalInvitationTournamentNITintheGardentodetermineanationalchampion.AlthoughpostseasontournamentshadoccurredbeforetheNITwasthefirstwithmajorcollegesfromdifferentregionsandprovedtobeagreatfinancialsuccess.TheNationalCollegiateAthleticAssociationNCAAcreateditsownpostseasontournamentin1939butdidnotrivaltheNITinprestigeforsometime.The1940ssawsignificantchangesforcollegebasketball.PlayersbeganusingthejumpshotafterKennySailorsofWyomingwowedtheEastwithitin
1943.Thebehind-the-backdribbleandpassalsoappearedasdidexceptionalbigmen.BobKurlandatOklahomaAMwasalmostsevenfeettallandGeorgeMikanatDePaulwassixfeetteninches.WhileKurlandhadperhapsthebettercollegecareerandplayedintwoOlympicshechosenottoplayprofessionalballwhereasMikanbecamethefirstdominantstarinthepros.Theirdefensiveplayinspiredtheruleagainstgoaltendingblockingashotonitsdownwardflight.AdolphRuppwhoplayedunderPhogAllenalsocoachedthefirstofhismanytalentedteamsatKentuckyinthatdecade.Howeverin1951Ruppandsixothercoachessufferedthroughapoint-shavingscandalthatinvolvedthirty-twoplayersatsevencollegesandseriouslyinjuredcollegebasketballparticularlyinNewYorkwherefourofthesevenschoolswerelocated.WhilethegamesurvivedtheNCAAmoveditstournamentawayfromMadisonSquareGardentodifferentcitieseachyearandtheNITsprestigebegantodecline.Professionalbasketballremainedadisorganizedandstodgysportupuntilthelate1940swithbarnstormingstillcentraltothegameandmostplayersstillusingthesetshot.In1946howeverhockeyownersledbyMauricePodoloffcreatedtheBasketballAssociationofAmericaBAAintheEasttofilltheirarenasbutfewfanscameevenafterJoeFulksofPhiladelphiaintroducedthejumpshot.TheBAAsrivaltheNationalBasketballLeaguehadexistedsincethe1930shadbetterplayerslikeMikanoftheMinneapolisLakersBobDaviesoftheRochesterRoyalsandDolphShayesoftheSyracuseNationalsbutoperatedinmuchworsefacilitiesanddidnotdomuchbetteratattractingaudiences.In1948PodoloffluredtheLakersRoyalsandtwootherteamstotheBAAandproposedamergerofthetwoleaguesforthe1949–1950season.TheresultwastheNationalBasketballAssociationNBAwithPodoloffitsfirstcommissioner.Theseventeen-teamleaguestruggledatfirstbutsoonreduceditssizeandgainedstabilityinlargepartbecauseofMikansappealandPodoloffsskills.Despitethepoint-shavingscandalcollegeballthrivedinthe1950slargelybecauseithadprolificscorersandmoregreatplayersthaninanypreviousdecade.FrankSelvyofFurmanandPaulArizinofVillanovabothaveragedoverfortypointsearlyinthedecadewhileClarenceBevoFrancisoftinyRioGrandeCollegeinOhioamazedfansbyscoring116pointsinonegamewhileaveraging50pergameforaseason.Thedecadealsowitnessedsomeofthemosttalentedandcompleteplayersever.TomGolaatLaSalleBillRussellatSanFranciscoWiltChamberlainatKansasElginBayloratSeattleJerryWestatWestVirginiaandOscarRobertsonatCincinnatiallhadphenomenalskillsthathavesincebeenthemeasureofotherplayers.Andin1960oneofthebestteamseverOhioStatewontheNCAAtitleledbyJerryLucasandJohnHavlicek.Professionalbasketballunderwentmajorchangesinthe1950sthathelpedincreaseitspopularity.In1950EarlLloydfromWestVirginiaplayedfortheWashingtonCapitolsandbecamethefirstAfricanAmericantoplayintheNBA.In1954DannyBiasoneowneroftheSyracuseNationalspersuadedtheNBAtoinstitutethetwenty-four-secondshotclockrequiringateamtoshootwithinthattime.ThiseliminatedtheslowpacethathadlongprevailedintheprosandmadetheNBAmoreexciting.Teamsnowscoredonehundredpointsagameregularly.Theleaguealsonowawardedfoulshotswhentheotherteamreceivedmorethanfivepersonalfoulsaperiodgreatlyreducingtheroughplaythathadhurttheprogame.In1956RedAuerbachoftheBostonCelticsmadethebestdealinNBAhistorywhenheacquiredthedraftrightstoBillRussellthedefensiveplayerandrebounderheneededtocomplementBobCousyandBillSharmaninthebackcourt.WiththeadditionofRusselltheCelticsbecamethebestproteameverwinningelevenofthenextthirteenchampionshiptitlesbeforeexpansiondilutedthetalentintheNBA.TheSt.LouisHawkswithBobPettitbeattheCelticsin1958andthePhiladelphia76erswithChamberlainbeatthemin
1967.ButRussellaplayer-coachfortwotitlesandhisteammatesformedthegreatestdynastyinproball.EventheLosAngelesLakerswhohadmovedfromMinneapolisin1960withWestandBaylorwerenomatchfortheCelticsovertheseyears.WhileWestBaylorChamberlain—whoaveragedoverfiftypointsagamein1962—andOscarRobertson—whointhesameyearaveragedatripledoublepergameinpointsassistsandrebounds—weresuperiortoanyindividualCelticnootherteamcouldconsistentlyplaydefensere-boundandrunwiththeCeltics.。