Chapter IIIElementaryFunctionsIn this chapter, we will generalize various elementary functions to corresponding functions of acomplex variable.To be specific, we define analytic functions of a complex variablez thatreduce to the elementary functions in calculus when z=x+io.We startby definingthecomplexexponentialfunctionandthenuseittodeveloptheotherss3.1.TheExponential Function1.PropertiesofexponentialfunctionsIf wewrite=x+iyandz2=x+iy2then=(ee)(e")=(ee)(e)=e+(+)=e+2eie'-2e2 =ee"i-=2e"2de"=e'dzIf e'=βei where β=e and Φ=y,then leier andArg(e")= y + 2nπ (n =0,±1,+2,.)2.ExampleExample1.Thereare values of zsuch thate" =-1(3.1.9)To find them, we write equation (3.1.9) as e*e = le*. Then, by Proposition 1.8.1, we havee=1andy=元+2元(n=0,±l+2...)Thus,x=O,and we find that
Chapter Ⅲ Elementary Functions In this chapter, we will generalize various elementary functions to corresponding functions of a complex variable. To be specific, we define analytic functions of a complex variable that reduce to the elementary functions in calculus when z = + ixz 0 . We start by defining the complex exponential function and then use it to develop the others. §3.1. The Exponential Function 1.Properties of exponential functions If we write 111 = + iyxz and 222 = + iyxz then ))(())(( . 21 2211 21 21 2121 21 iyxiyxzz iyiyxx )( zzyyixx eeeeeeeeeeeee ++ + = = = = 221 1zzzz = eee − or 21 2 1 zz z z e e e − = . zz ee dz d = If where and φ β iz = ee x β = e φ = y ,then and xz || = ee nnye ±±=π+= K),2,1,0(2)(Arg z . 2. Example Example 1. There are values of z such that −= 1 z e . (3.1.9) To find them, we write equation (3.1.9) as . Then, by Proposition 1.8.1, we have iiyx π = 1eee = 1 x e and = π + π nny = ± ± K),2,1,0(2 . Thus, x = 0, and we find that
s3.2.TheLogarithmicFunction1.Definition of logarithm of a complex numberIf wsatisfied(3.2.1)o=where z is anynonzero complex number,then w is calleda logarithm ofthe number z2.Definition of a logarithmic functionThe setLogz = (lnz|+i(argz +2n元): neZ)=ln+iArgz, Vz eCl (0) (3.2.2)is called the logarithm of z.Usually,we writeLogz = In|z|+i(argz+2n)(n = 0, ±1, ±2,.),and then get a simple relationelg=(0)(3.2.3)Thus, we get a multi-valued functionLog :CI(0) →C,called the logarithmic function.3.ExamplesExample1.If z =-1-3i, then r = 2 and 0=-2元/3.Hence1_2+2n元)=1n2+2|nLog(-1- /3i) = In2 +i(n=0,±1,±2...)33Equality (3.2.3)isvalidforall nonzerocomplexnumber,but theequality Loge=zisnottrue.To find Loge=z,we use the definition of e'(Sec.3.1)and see that[e=e* and Arg(e")=y+2n (n=0, ±1,±2,.)when z= x+iy.Hence, we know thatLog(e')=Inlei+iArg(e')= In(e*)+i(y+2nπ)=(x+iy)+2nmi (n=0, ±1, ±2,..)ThereforeLog(ei)= z+2nπi (n=0, ±1, ±2,..)(3.2.4)The principal value of Logz is the value obtained from equation (3.2.2) when n =0there and is denoted by log z. Thuslogz = Inr+iargz(3.2.5)Note that logz is well defined and single-valued when z + O and thatLogz=logz+2nπi(n=0,±1±2,...)(3.2.6)Clearly, logz reduces to the usual logarithm in calculus when z is a positive real numberz=r.To see this, one need only write z=reio, in which case equation (3.2.5)becomeslogz=Inr,That is, logr=Inr.Example 2. From expression (3.2.2), we find thatLogl=lnl+i(0+2n元)=2nπi(n=0,±1,±2,...)As expected, log1=0Our final example here reminds us that, although we were unable to find logarithms ofnegative real numbers in calculus,we can now do soExample 3. Observe thatLog(-1)=ln1+i(π+2n元)=(2n+1)πi(n=0,±1,±2,.)and that log(-1)= πi
§3.2. The Logarithmic Function 1.Definition of logarithm of a complex number If w satisfied zew = (3.2.1) where z is any nonzero complex number,then w is called a logarithm of the number z . 2. Definition of a logarithmic function The set { } π Z zziznnzizz ∈∀+=∈++= C }0{\,Argln:)2(arg||lnLog (3.2.2) is called the logarithm of z . Usually, we write = + + nzizz π )2(arg||lnLog n = ± ± K),2,1,0( , and then get a simple relation }{ Log ze z = z ≠ )0( . (3.2.3) Thus, we get a multi-valued function }0{\:Log → CC , called the logarithmic function. 3. Examples Example 1. If −−= 31 iz , then r = 2 and θ = − π 3/2 . Hence nii π πin π ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ ⎟ −+=⎠ ⎞ ⎜ ⎝ ⎛ +−+=−− 3 1 22ln2 3 2 2ln)31(Log n = ±± K),2,1,0( . Equality (3.2.3) is valid for all nonzero complex number, but the equality is not true. To find , we use the definition of (Sec. 3.1) and see that zez Log = zez Log = z e xz = ee and nye π+= z 2)(Arg n = ± ± K),2,1,0( when z x += iy . Hence, we know that iniyxnyieeieez z z x π ++=++=+= 2)()2()ln()(Argln)(Log π n ±±= K),2,1,0( . Therefore, inzez += 2)(Log π n = ± ± K),2,1,0( . (3.2.4) The principal value of is the value obtained from equation (3.2.2) when there and is denoted by . Thus Logz n = 0 log z = + arglnlog zirz . (3.2.5) Note that log z is well defined and single-valued when z ≠ 0 and that = + 2logLog πinzz n = ± ± K),21,0( . (3.2.6) Clearly, log z reduces to the usual logarithm in calculus when z is a positive real number z = r . To see this, one need only write , in which case equation (3.2.5) becomes . That is, . i0 = rez = lnlog rz = lnlog rr Example 2. From expression (3.2.2), we find that += + π = 2)20(1ln1Log πinni n = ± ± K),2,1,0( . As expected, = 01log . Our final example here reminds us that, although we were unable to find logarithms of negative real numbers in calculus, we can now do so. Example 3. Observe that − += π + π = + )12()2(1ln)1(Log πinni n = ± ± K),2,1,0( and that )1log( =− πi
s3.3.BranchesandDerivativesofLogarithms1. Properties of the branche Lα(z)PutD.=(reie:r>0,α<<α+2元),a function L.D→C defined byLα(z)= lnr+i0(z=rei,r>0,α<0<α+2元).(3.3.2)From this definition, we can prove that the function La(2) has the following properties.(1) ela() =z(Vze Da);(2) f,(z)= L(2n-1)x(z), VneZ, whenever -π<argz<π ;(3) Logz =(L(=):αER)(VzECI(0));(4) La(Da)= ((u,v): ueR,α<v<α+2元) ;L(-)=1(=ren,>0,α<0<α+2),(5)dzZ2.Definition of a branch of a multi-valued functionA branch of a multi-valued function F defined on D is any single-valued functionf:E→C suchthat(I) f is analytic on the domain E;(2) EcD; and(3) VzeE, f(z)eF(z)
§3.3. Branches and Derivatives of Logarithms 1. Properties of the branche α z)(L Put ,0:{ παθα }2 , a function defined by θ α rreD +<<>= i : DL αα → C L ) α = ln)( + irz θ ,0,( 2παθα θ rrez +<<>= i . (3.3.2) From this definition, we can prove that the function zL )( has the following properties. α (1) )( ; )( α α Dzze zL ∈∀= (2) = ∈∀ Z , whenever − nzLzfn n ),()( )12( π − π < arg z < π ; (3) = α zLz α ∀∈ z ∈CR })0{\}(:)({Log ; (4) αα = ∈R,:),{()( α < vuvuDL < α + π}2 ; (5) z z dz d 1 α )(L = ,0,( παθα )2 θ rrez +<<>= i . , 2. Definition of a branch of a multi-valued function A branch of a multi-valued function defined on is any single-valued function such that F D : Ef → C (1) f is analytic on the domain E ; (2) E ⊂ D ; and (3) z ∈∀ E , ∈ zFzf )()(
s3.4.SomeIdentitiesonLogarithmsAs suggested by relations (3.2.3) in Sec.3.2, as well as Exercises 3, 4, and 5 with Sec.3.3, someidentities involving logarithms in calculus carry over to complex analysis and others do not. In thissection, wederive a fewthat do carry over, sometimes with qualifications as to how they are to beinterpreted. A reader who wishes to pass to Sec. 3.2 can simply refer to results here when needed.1.Operations of LogzIf z, and z2 denote any two nonzero complex numbers, then(3.4.2)Log(z,-2)= Log +Log2, Vz,2 0Log = Log-, - Log2(3.4.3)Z22. Properties of LogzWeincludetwootherpropertiesof Logzthatwillbeofspecial interestinz"=emLog(n=0,±1,±2,...)(3.4.4)When n =1, this reduces, of course, to relation (3.2.3), Sec. 3.2. Equation (3.4.4) is readilyverified by writing z = reie and noting that each side becomes r"eine.Also,21/" =exp(-Logz) (n=1, 2,..)(3.4.5)nThat is, the term on the right here has n distinct values, and those values are the nth roots ofz.Toprovethis,wewrite z=rexp(i),where istheprincipal valueof Argz.Then,inview of definition (3.2.2), Sec.3.2, of Logz,(1(0+2k元)exp(Logz)=^expl-Inr+kETnn(nThus,fromthedefinitionoftheexponential function,weobtain that(.0+2k元）1/n:keZ(3.4.6)exp(=Logz)=3"/rexplnnThis establishes property (3.4.5), which is also valid for every negative integer n too (seeExercise 5)
§3.4. Some Identities on Logarithms As suggested by relations (3.2.3) in Sec. 3.2, as well as Exercises 3, 4, and 5 with Sec. 3.3, some identities involving logarithms in calculus carry over to complex analysis and others do not. In this section, we derive a few that do carry over, sometimes with qualifications as to how they are to be interpreted. A reader who wishes to pass to Sec. 3.2 can simply refer to results here when needed. 1.Operations of Logz If and denote any two nonzero complex numbers, then 1 z 2 z 21 1 LogLog)(Log 2 = + zzzz , 0 ∀ zz 21 ≠ . (3.4.2) Log ,LogLog 1 2 2 1 zz z z −= (3.4.3) 2. Properties of Logz We include two other properties of Logz that will be of special interest in ),2,1,0( = Logznn nez ±±= K . (3.4.4) When , this reduces, of course, to relation (3.2.3), Sec. 3.2. Equation (3.4.4) is readily verified by writing and noting that each side becomes . Also, n = 1 iθ = rez inn θ er ),2,1()Log 1 exp( /1 = nz = K n z n (3.4.5) That is, the term on the right here has distinct values, and those values are the roots of . To prove this, we write n nth z = irz θ )exp( , where θ is the principal value of . Then, in view of definition (3.2.2), Sec. 3.2, of , Argz Logz ⎭ ⎬ ⎫ ⎩ ⎨ ⎧ ⎟ ∈ ⎠ ⎞ ⎜ ⎝ ⎛ + = + k Z n ki r n z n : )2( ln 1 exp)Log 1 exp( πθ . Thus, from the definition of the exponential function, we obtain that : . 2 exp)Log 1 exp( n /1 n zk n k irz n = ⎭ ⎬ ⎫ ⎩ ⎨ ⎧ ⎟ ∈ ⎠ ⎞ ⎜ ⎝ ⎛ + = Z πθ (3.4.6) This establishes property (3.4.5), which is also valid for every negative integer too (see Exercise 5). n
s3.5.ComplexPowerFunctions1.Definitionof acomplexpowerfunctionWhen cisanycomplexnumber,thecomplexpowerzofanonzerocomplexnumberzisdefinedbymeansoftheequation- =eclog, z+0(3.5.1)Thus,we obtain amultiple-valuedfunction w= z(z+O),calleda complexpowerfunction2.ExamplesExample 1. Powers of z are, in general, multiple-valued, as illustrated by writing7-2i = exp(-2iLogi)and then1Logi = In1 +2n+(n =0, ±1, ±2,...)+2n元22This shows that1-2 = exp[(4n+1)元](n = 0, ±1, ±2,.) .(3.5.2)Note that these values of i-2iiareall realnumbers.Since the exponential function has the property 1/e=e-", one can see that11= exp(-cLog-)= 2-czcexp(cLog)and, in particular, that 1/i2 = i-2i. According to expression (3.5.2), then,(3.5.3)=exp[(4n+ )](n =0, 1, 2..),.The principal value of z° occurs when Logz is replaced by log z in definition (3.5.1):P.V. 2 = elog= =(-')-x(3.5.5)Example 2. The principal value of (-i)’isexp[ilog(-i)] = expl / In1.expThat is,元P.V.(-i)" = exp(3.5.6)Example 3. The principal branch of ≥2/3 can be witten(2222argz=3/r2exp-lnr+_iarg=expllogz=exp333.3ThusP.V./=cos2ag+Fsin,2argz(3.5.7)33w,asonecanseedirectlyfromTheorem2.12.1This function is analytic in thedomainD
§3.5. Complex Power Functions 1.Definition of a complex power function When is any complex number, the complex power of a nonzero complex number is defined by means of the equation c c z z zcc ez Log = , z ≠ 0. (3.5.1) Thus, we obtain a multiple-valued function zzw ≠= )0( , called a complex power function. c 2. Examples Example 1. Powers of z are, in general, multiple-valued, as illustrated by writing )Log2exp( 2 i ii i −= − and then ),2,1,0( 2 1 22 2 1lnLog ⎟ ±±= K ⎠ ⎞ ⎜ ⎝ ⎛ ⎟ +=⎠ ⎞ ⎜ ⎝ ⎛ ++= π π nninii π . This shows that ),2,1,0]()14exp[( i −2i π nn ±±=+= K . (3.5.2) Note that these values of are all real numbers. i i −2 Since the exponential function has the property , one can see that zz ee − /1 = c c zzc zcz − = )Logexp( =−=)Logexp( 11 and, in particular, that . According to expression (3.5.2), then, ii ii 22 /1 − = ).,2,1,0]()14exp[( 1 2 nn ±±=+= K i i π (3.5.3) The principal value of occurs when is replaced by in definition (3.5.1): c z Logz log z == −π .P.V )( log czcc zez . (3.5.5) Example 2. The principal value of is i −i)( 2 exp 2 1lnexp)]log(exp[ ππ =⎥ ⎦ ⎤ ⎢ ⎣ ⎡ ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ =− − iiii . That is, . 2 exp)(.P.V π =− i i (3.5.6) Example 3. The principal branch of can be written 3/2 z . 3 arg2 arg exp 3 2 ln 3 2 explog 3 2 exp 3 2 ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ ⎟ = ⎠ ⎞ ⎜ ⎝ ⎛ ⎟ = + ⎠ ⎞ ⎜ ⎝ ⎛ z z irzir Thus . 3 arg2 sin 3 arg2 .P.V cos 3/2 3 2 3 2 z ri z = rz + (3.5.7) This function is analytic in the domain , as one can see directly from Theorem 2.12.1. D π−