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The complex logarithm, exponential and power functions

Physics 116 AWinter 2011 The complex logarithm, exponential and power functionsIn these notes, we examine the logarithm, exponential and power functions , wherethe arguments of these functions can be complex numbers. In particular, we areinterested in how their properties differ from the properties of thecorrespondingreal-valued functions . 1. Review of the properties of the argument of a complex numberBefore we begin, I shall review the properties of the argument of anon-zerocomplex numberz, denoted by argz(which is a multi-valued function), and theprincipal valueof the argument, Argz, which is single-valued and conventionallydefined such that: <Argz .(1)Details can be found in the class handout entitled,The argument of a complexnumber.

where the integer Nn is given by: Nn = 1 2 − n 2π Arg z , (16) and [ ] is the greatest integer bracket function introduced in eq. (4). 2. Properties of the real-valued logarithm, exponential and power func-

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Transcription of The complex logarithm, exponential and power functions

1 Physics 116 AWinter 2011 The complex logarithm, exponential and power functionsIn these notes, we examine the logarithm, exponential and power functions , wherethe arguments of these functions can be complex numbers. In particular, we areinterested in how their properties differ from the properties of thecorrespondingreal-valued functions . 1. Review of the properties of the argument of a complex numberBefore we begin, I shall review the properties of the argument of anon-zerocomplex numberz, denoted by argz(which is a multi-valued function), and theprincipal valueof the argument, Argz, which is single-valued and conventionallydefined such that: <Argz .(1)Details can be found in the class handout entitled,The argument of a complexnumber.

2 Here, we recall a number of results from that handout. One can regardargzas a set consisting of the following elements,argz= Argz+ 2 n , n= 0, 1, 2, 3, .. , <Argz .(2)One can also express Argzin terms of argzas follows:Argz= argz+ 2 12 argz2 ,(3)where [ ] denotes the greatest integer function. That is, [x] is defined to be thelargest integer less than or equal to the real numberx. Consequently, [x] is theunique integer that satisfies the inequalityx 1<[x] x ,for realxand integer [x].(4) Note that the wordargumenthas two distinct meanings. In this context, given a functionw=f(z), we say thatzis the argument of the functionf. This should not be confused withthe argument of a complex number, argz. The following three books were particularly useful in the preparation of these Variables and Applications, by James Ward Brown and Ruel V.

3 Churchill (McGrawHill, New York, 2004). of complex Variables, by Louis L. Pennisi, with the collaboration of Louis I. Gordonand Sim Lasher (Holt, Rinehart and Winston, New York, 1963). Theory of Analytic functions : A Brief Course, by Markushevich (Mir Publishers,Moscow, 1983).1 For example, [ ] = [1] = 1 and [ ] = 1. One can check that Argzasdefined in eq. (3) does fall inside the principal interval specified by eq. (1).The multi-valued function argzsatisfies the following properties,arg(z1z2) = argz1+ argz2,(5)arg z1z2 = argz1 argz2.(6)arg 1z = argz= argz .(7)Eqs. (5) (7) should be viewed as set equalities, the elements of the sets indi-cated by the left-hand side and right-hand side of the above identities , the following results arenotset equalities:argz+ argz6= 2 argz ,argz argz6= 0,(8)which, by virtue of eqs.

4 (5) and (6), yield:argz2= argz+ argz6= 2 argz ,arg(1) = argz argz6= 0.(9)For example, arg(1) = 2 n, forn= 0 1, 2, .. More generally,argzn= argz+ argz+ argz|{z}n6=nargz .(10)We also note some properties of the the principal value of the (z1z2) = Argz1+ Argz2+ 2 N+,(11)Arg (z1/z2) = Argz1 Argz2+ 2 N ,(12)where the integersN are determined as follows:N = 1,if Argz1 Argz2> ,0,if <Argz1 Argz2 ,1,if Argz1 Argz2 .(13)If we setz1= 1 in eq. (12), we find thatArg(1/z) = Argz=(Argz ,if Imz= 0 andz6= 0, Argz ,if Imz6= 0.(14)Note that forzreal, both 1/zandzare also real so that in this casez=zandArg(1/z) = Argz= Argz. In addition, in contrast to eq. (10), we haveArg(zn) =nArgz+ 2 Nn,(15)2where the integerNnis given by:Nn= 12 n2 Argz ,(16)and [ ] is the greatest integer bracket function introduced in eq.)

5 (4).2. Properties of the real-valued logarithm, exponential and power func-tionsConsider the logarithm of a positive real number. This function satisfies anumber of properties:elnx=x ,(17)ln(ea) =a ,(18)ln(xy) = ln(x) + ln(y),(19)ln xy = ln(x) ln(y),(20)ln 1x = ln(x),(21)lnxp=plnx ,(22)for positive real numbersxandyand arbitrary real numbersaandp. Likewise,the power function defined over the real numbers satisfies:xa=ealnx,(23)xaxb=xa+b,(24)xax b=xa b,(25)1xa=x a,(26)(xa)b=xab,(27)(xy)a=xaya,(28) xy a=xay a,(29)for positive real numbersxandyand arbitrary real numbersaandb. Closelyrelated to the power function is the generalized exponential function defined over3the real numbers. This function satisfies:ax=exlna,(30)axay=ax+y,(31)axa y=ax y,(32)1ax=a x,(33)(ax)y=axy,(34)(ab)x=axbx,(35) ab x=axb x.

6 (36)for positive real numbersaandband arbitrary real would like to know which of these relations are satisfied when thesefunc-tions are extended to the complex plane. It is dangerous to assumethat all ofthe above relations are valid in the complex plane without modification,as thisassumption can lead to seemingly paradoxical conclusions. Here arethree exam-ples:1. Since 1/( 1) = ( 1)/1 = 1,r1 1=1i=r 11=i1.(37)Hence, 1/i=iori2= 1. Buti2= 1, so we have proven that 1 = Since 1 = ( 1)( 1),1 = 1 =p( 1)( 1) = ( 1)( 1) =i i= 1.(38)3. To prove that ln( z) = ln(z) for allz6= 0, we proceed as follows:ln(z2) = ln[( z)2],ln(z) + ln(z) = ln( z) + ln( z),2 ln(z) = 2 ln( z),ln(z) = ln( z).Of course, all these proofs are faulty.

7 The fallacy in the first two proofs canbe traced back to eqs. (28) and (29), which are true for real-valued functions butnot true in general for complex -valued functions . The fallacy in thethird proofis more subtle, and will be addressed later in these notes. A carefulstudy of thecomplex logarithm, power and exponential functions will reveal howto correctlymodify eqs. (17) (36) and avoid pitfalls that can lead to false Definition of the complex exponential functionWe begin with the complex exponential function, which is defined via itspowerseries:ez= Xn=0znn!,wherezis any complex number. Using this power series definition, one can verifythat:ez1+z2=ez1ez2,for all complexz1andz2.(39)In particular, ifz=x+iywherexandyare real, then it follows thatez=ex+iy=exeiy=ex(cosy+isiny).

8 One can quickly verify that eqs. (30) (33) are satisfied by the complex exponentialfunction. In addition, eq. (34) clearly holds when the outer exponent is an integer:(ez)n=enz,n= 0, 1, 2, ..(40)If the outer exponent is a non-integer, then the resulting expression is a multi-valued power function. We will discuss this case in more detail in section moving on, we record one key property of the complex exponential :e2 in= 1,n= 0, 1, 2, 3, ..(41)4. Definition of the complex logarithmIn order to define the complex logarithm, one must solve the complexequation:z=ew,(42)forw, wherezis any non-zero complex number. If we writew=u+iv, theneq. (42) can be written aseueiv=|z|eiargz.(43)Eq. (43) implies that:|z|=eu,v= argz.

9 The equation|z|=euis a real equation, so we can writeu= ln|z|, where ln|z|isthe ordinary logarithm evaluated with positive real number arguments. Thus,w=u+iv= ln|z|+iargz= ln|z|+i(Argz+ 2 n), n= 0, 1, 2, 3, ..(44)5We callwthe complex logarithm and writew= lnz. This is a somewhat awkwardnotation since in eq. (44) we have already used the symbol ln for thereal shall finesse this notational quandary by denoting the real logarithm in eq. (44)by the symbol Ln. That is, Ln|z|shall denote the ordinary real logarithm of|z|.With this notational convention, we rewrite eq. (44) as:lnz= Ln|z|+iargz= Ln|z|+i(Argz+ 2 n), n= 0, 1, 2, 3, ..(45)for any non-zero complex , lnzis a multi-valued function (as its value depends on the integern).

10 It is useful to define a single-valuedcomplexfunction, Lnz, called the principalvalue of lnzas follows:Lnz= Ln|z|+iArgz , <Argz ,(46)which extends the definition of Lnzto the entire complex plane (excluding theorigin,z= 0, where the logarithmic function is singular). In particular, eq. (46)implies that Ln( 1) =i . Note that for real positivez, we have Argz= 0, sothat eq. (46) simply reduces to the usual real logarithmic functionin this relation between lnzand its principal value is simple:lnz= Lnz+ 2 in , n= 0, 1, 2, 3, ..5. Properties of the complex logarithmWe now consider which of the properties given in eqs. (17) (22) apply to thecomplex logarithm. Since we have defined the multi-value function lnzand thesingle-valued function Lnz, we should examine the properties of both these func-tions.


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