4 Heavisides Method This practical method was popularized by the English electrical engineer Oliver Heaviside 18501925 A typical application of the method is to solve 1 1 for the expression cos sin T ID: 44188
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5.4Heaviside'sMethod231 MultipleRoots.Assume(1)hasrealcoecientsandthedenomi-natorofthefraction(1)haspossiblymultipleroots.LetNpbethemultiplicityofrealrootspandletMqbethemultiplicityofcomplexroot q+iq,1pN,1qM.Thepartialfractionexpansionof(1)isgivenintermsofrealconstantsAp;k,Bq;k,Cq;kbyNXp=1X1kNpAp;k (s�sp)k+MXq=1X1kMqBq;k+Cq;k(s� q) ((s� q)2+2q)k:(5)AFailsafeMethodConsidertheexpansioninpartialfractionss�1 s(s+1)2(s2+1)=A s+B s+1+C (s+1)2+Ds+E s2+1:(6)TheveundeterminedrealconstantsAthroughEarefoundbyclearingthefractions,thatis,multiply(6)bythedenominatoronthelefttoobtainthepolynomialequations�1=A(s+1)2(s2+1)+Bs(s+1)(s2+1)+Cs(s2+1)+(Ds+E)s(s+1)2:(7)Next,vedi erentvaluesofsaresubstitutedinto(7)toobtainequationsfortheveunknownsAthroughE.Wealwaysusetherootsofthedenominatortostart:s=0,s=�1,s=i,s=�iaretherootsofs(s+1)2(s2+1)=0.Eachcomplexrootresultsintwoequations,bytakingrealandimaginaryparts.Thecomplexconjugateroots=�iisnotused,becauseitduplicatestheexistingequationobtainedfroms=i.Thethreerootss=0,s=�1,s=igiveonlyfourequations,sos=1isusedtogetthefthequation:�1=A(s=0)�2=�2C�2(�D+E)(s=�1)i�1=(Di+E)i(i+1)2(s=i)0=8A+4B+2C+4(D+E)(s=1)(8)BecauseDandEarereal,thecomplexequation(s=i)becomestwoequations,asfollows.i�1=(Di+E)i(i2+2i+1)Expandpower.i�1=�2Di�2ESimplifyusingi2=�1.1=�2DEquateimaginaryparts.�1=�2EEquaterealparts.Solvingthe55system,theanswersareA=�1,B=2,C=0,D=�1=2,E=1=2. 5.4Heaviside'sMethod233 ExtensiontoMultipleRoots.AnextensionofHeaviside'smethodispossibleforthecaseofrepeatedroots.Thebasicideaistofactor{outtherepeats.Toillustrate,considerthepartialfractionexpansiondetailsR=1 (s+1)2(s+2)Asamplerationalfunctionhavingrepeatedroots.=1 s+11 (s+1)(s+2)Factor{outtherepeats.=1 s+11 s+1+�1 s+2Applythecover{upmethodtothesimplerootfraction.=1 (s+1)2+�1 (s+1)(s+2)Multiply.=1 (s+1)2+�1 s+1+1 s+2Applythecover{upmethodtothelastfractionontheright.Termswithonlyonerootinthedenominatorarealreadypartialfrac-tions.Thustheworkcentersonexpansionofquotientsinwhichthedenominatorhastwoormoreroots.SpecialMethods.Heaviside'smethodhasausefulextensionforthecaseofrootsofmultiplicitytwo.Toillustrate,considerthesedetails:R=1 (s+1)2(s+2)Afractionwithmultipleroots.=A s+1+B (s+1)2+C s+2Seeequation(5).=A s+1+1 (s+1)2+1 s+2FindBandCbyHeaviside'scover{upmethod.=�1 s+1+1 (s+1)2+1 s+2Multiplybys+1.Sets=1.Then0=A+1.Theillustrationworksforonerootofmultiplicitytwo,becauses=1willresolvethecoecientnotfoundbythecover{upmethod.Ingeneral,ifthedenominatorin(1)hasaroots0ofmultiplicityk,thenthepartialfractionexpansioncontainstermsA1 s�s0+A2 (s�s0)2++Ak (s�s0)k:Heaviside'scover{upmethoddirectlyndsAk,butnotA1toAk�1.