math132-complexnumbersandfunctions数学132复杂的数字和功能

《math132-complexnumbersandfunctions数学132复杂的数字和功能》由会员上传分享，免费在线阅读，更多相关内容在教育资源-天天文库。

1、Math132-ComplexNumbersandFunctionsManyengineeringproblemscanbetreatedandsolvedbyusingcomplexnumbersandcomplexfunctions.Wewilllookatcomplexnumbers,complexfunctionsandcomplexdifferentiation.PartIComplexNumbersInalgebrawediscoveredthatmanyequationsarenotsatisfiedbyanyrealnumb

2、ers.Examplesare:orWemustintroducetheconceptofcomplexnumbers.Definition:Acomplexnumberisanorderedpairofrealnumbersxandy.Wecallxtherealpartofzandytheimaginarypart,andwewrite,.Example1:andTwocomplexnumbersareequalwhereand:ifandonlyifandAdditionandSubtractionofComplexNumbers:W

3、edefinefortwocomplexnumbers,thesumanddifferenceofand:and.Multiplicationoftwocomplexnumbersisdefinedasfollows:Example2:Letandthenandand.Weneedtorepresentcomplexnumbersinamannerthatwillmakeadditionandmultiplicationeasiertodo.13Math132-ComplexNumbersandFunctionsComplexnumbers

4、representedasAcomplexnumberwhoseimaginarypartis0isoftheformandwehaveandandwhichlookslikerealaddition,subtractionandmultiplication.Soweidentifywiththerealnumberandthereforewecanconsidertherealnumbersasasubsetofthecomplexnumbers.Welettheletterandwecalliapurelyimaginarynumber

5、.Nowconsiderandsowecanconsiderthecomplexnumber=therealnumber.WealsogetAndsowehave:Nowwecanwriteadditionandmultiplicationasfollows:Example3:Letand,thenandand=.TheComplexPlaneThegeometricrepresentationofcomplexnumbersistorepresentthecomplexnumberasthepoint.y-axis2112x-axis13

6、Math132-ComplexNumbersandFunctionsSotherealnumberisthepointonthehorizontalx-axis,thepurelyimaginarynumberisontheverticaly-axis.Forthecomplexnumber,istherealpartandyistheimaginarypart.Example4.Locate2-3ionthegraphabove.Howdowedividecomplexnumbers?Let’sintroducetheconjugateo

7、facomplexnumberthengotodivision.Giventhecomplexnumber,definetheconjugateWecandividebyusingthefollowing:Example5.ProblemSetIFind1.2.3.4.5.6.7.8.9.Letandand,find10.11.12.13.14.15.Graphthefollowing:16.17.18.anditsconjugate.19.Findthesolutionsof.20.Findthesolutionof.ComplexNum

8、bersinPolarFormItispossibletoexpresscomplexnumbersinpolarform.Ifthepointisrepresentedbypo