complex_variables--James Nearing_university Miami.pdf

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1、ComplexVariablesInthecalculusoffunctionsofacomplexvariabletherearethreefundamentaltools,thesamefunda-mentaltoolsasforrealvariables.Dierentiation,Integration,andPowerSeries.I'llrstintroduceallthreeinthecontextofcomplexvariables,thenshowtherelationsbetweenthem.Theapplicationsofthe

2、subjectwillformthemajorpartofthechapter.14.1DierentiationWhenyoutrytodierentiateacontinuousfunctionisitalwaysdierentiable?Ifit'sdierentiableonceisitdierentiableagain?Theanswertobothisno.Takethesimpleabsolutevaluefunctionoftherealvariablex.x(x0)f(x)=jxj=x(x<0)Thishasaderiva

3、tiveforallxexceptzero.Thelimit8<1(x>0)f(x+
x)f(x)!1(x<0)(14:1)
x:?(x=0)worksforbothx>0andx<0.Ifx=0however,yougetadierentresultdependingonwhether
x!0throughpositiveorthroughnegativevalues.Ifyouintegratethisfunction,Zx00x2=2(x0)jxjdx=20x=2(x<0)theresulthasaderivativeeverywher

4、e,includingtheorigin,butyoucan'tdierentiateittwice.Afewmoreintegrationsandyoucanproduceafunctionthatyoucandierentiate42timesbutnot43.Therearefunctionsthatarecontinuousbutwithnoderivativeanywhere.They'reharder*toconstruct,butifyougranttheirexistencethenyoucanrepeattheprecedingman

5、ipulationandcreateafunctionwithanynumberofderivativeseverywhere,butnomorethanthatnumberanywhere.Foraderivativetoexistatapoint,thelimitEq.(14.1)musthavethesamevaluewhetheryoutakethelimitfromtherightorfromtheleft.Extendtheideaofdierentiationtocomplex-valuedfunctionsofcomplexvariabl

6、es.Justchangetheletterxtotheletterz=x+iy.Examineafunctionsuchasf(z)=z2=x2y2+2ixyorcosz=cosxcoshy+isinxsinhy.Canyoudierentiatethese(yes)andwhatdoesthatmean?0f(z+
z)f(z)dff(z)=lim=(14:2)
z!0
zdzistheappropriatedenition,butforittoexistthereareevenmorerestrictionsthanintherealcase

7、.Forrealfunctionsyouhavetogetthesamelimitas
x!0whetheryoutakethelimitfromtherightP1kk*Weierstrasssurprisedtheworldofmathematicswithacos(bx).Ifa<1whileab>10thisiscontinuousbuthasnoderivativeanywhere.Thisstatementismuchmorediculttoprovethanitlooks.JamesNearing,UniversityofMiami114

8、

9、ComplexVariables2orfromtheleft.In