- 33 -Chapter IIAnalytic FunctionsWe now discuss complex-valued functions of a complex variable and develop atheory of differentiation for them.The main goal of the chapter is to introduceanalyticfunctions,whichplay a central role in complex analysis.S2.1.FunctionsofaComplex Variable1.Definition of complex-valued functions of a complex variableLet D be a set of complex numbers. Afiunction f defined on D is a rule thatassigns to each z in D a complex number w.The number w is called thevalue of f at z and is denoted by f(=); that is, w = f(z). The set D iscalled the domain of definition of f and set f(D) is called the range of fSince the variable = and the value f() of a function f at = are allcomplex numbers, we call such a function a complex-valued functions ofa complexvariable.2.ExamplesExample1.Thefunctionw=1/zisdefinedonthesetD=zEC:z+0For a complex-valued function f of complex variable z=x+iy definedon D,putu(x,y)= Re f(x + iy),v(x,y)=Im f(x +iy),then we obtain two real-valued functions u andydefined on D so thatw = f(z) can be expressed in terms of a pair of real-valued functions of realvariablesxandy:W=f(=)=u(x,y)+iv(x,y), Vz=x+iyeD(2.1.1)Ifthe polar coordinates r and ,instead of x and y,are used, thenw= f(=)= u(r,0)+iv(r,0), Vz = re" e D,(2.1.2)where u(r,0)= Re f(re),v(r,0)= Im f(re)
- 33 - Chapter Ⅱ Analytic Functions We now discuss complex-valued functions of a complex variable and develop a theory of differentiation for them. The main goal of the chapter is to introduce analytic functions, which play a central role in complex analysis. §2.1. Functions of a Complex Variable 1.Definition of complex-valued functions of a complex variable Let be a set of complex numbers. A function defined on is a rule that assigns to each D f D z in a complex number . The number is called the value of at and is denoted by ; that is, D w w f z zf )( = zfw )( . The set is called the domain of definition of and set is called the range of . Since the variable and the value of a function at are all complex numbers, we call such a function a complex-valued functions of a complex variable. D f Df )( f z zf )( f z 2.Examples Example 1. The function = /1 zw is defined on the set C zzD ≠∈= }0:{ . For a complex-valued function f of complex variable z = x + iy defined on D, put = + = + iyxfyxviyxfyxu )(Im),(),(Re),( , then we obtain two real-valued functions u and defined on so that can be expressed in terms of a pair of real-valued functions of real variables v D = zfw )( x and y : = = + ),(),()( ∀ = + ∈ Diyxzyxivyxuzfw . (2.1.1) If the polar coordinates r and θ , instead of x and y , are used, then Drezrivruzfw , (2.1.2) i ∈=∀θ+θ== θ ),(),()( where )(Im),(),(Re),( . θ θ =θ =θ i i refrvrefru
Chapter II- 34 -Analytic FunctionsExample 2. If f() = z2, thenf(x+iy)=(x+iy)? = x2 - y? +i2xyHenceu(x,y)=x2-y? and v(x,y)=2xyWhen polar coordinates are used,f(rei0)=(rei0)? = r2ei20 = r? cos20+ir? sin20Consequently,u(r,0)=r2 cos20 and v(r,0)=r? sin20If, in either of equations (2.1.1) and (2.1.2), the function always has valuezero, then the value of f is always real. That is, f is a real-valued function ofacomplexvariableExample 3. A real-valued function that is used to illustrate some importantconcepts later in this chapter isW=f()=P=x+y +i0If n is zero or a positive integer and if ao.aj,a2...,a., are complexconstants with a, +O,then the functionw=P(z)=ao+az+a2z+..+a,-"is called a polynomial of degree n.Note that the sum here has a finite number ofterms andthatthedomain ofdefinition is theentirez-plane.A quotientP(z)/Q(z)ofpolynomials is calleda rational fumnction and aredefined at eachpointzwhere Q(z)0:Polynomials and rational functions constituteelementary,but important,classes offunctions ofa complexvariable.Ageneralization of the concept of function, called a multiple-valued functionis a rule that assigns more than one value to a point z in the domain of definitionThesemultiple-valuedfunctionsoccurinthetheoryoffunctionsofa complexvariable,justastheydointhecaseofreal variables.Whenmultiple-valuedfunctions are studied, usually just one of the possible values assigned to each pointistaken,ina systematicmanner,anda single-valued function is constructedfromthe multi-valued function.Example4.Letzdenoteanynonzero complexnumber.Weknowfrom Sec.1.8 that 21/2 has the two values:
Chapter Ⅱ Analytic Functions - 34 - Example 2. If , then 2 )( = zzf )()( 2xyiyxiyxiyxf 222 +−=+=+ . Hence 22 ),( −= yxyxu and = 2),( xyyxv . When polar coordinates are used, θ θ θ θ θ )()( 2sin2cos 2222 2 rerreref ir i i i === + . Consequently, θ 2cos),( θ 2 = rru and θ 2sin),( θ . 2 = rrv If, in either of equations (2.1.1) and (2.1.2), the function always has value zero, then the value of is always real. That is, is a real-valued function of a complex variable. v f f Example 3. A real-valued function that is used to illustrate some important concepts later in this chapter is ||)( .0 222 ++=== iyxzzfw If is zero or a positive integer and if are complex constants with , then the function n aaaa n , 210 K an ≠ 0 n n L++++== zazazaazPw 2 210 )( is called a polynomial of degree . Note that the sum here has a finite number of terms and that the domain of definition is the entire -plane. A quotient of polynomials is called a rational function and are defined at each point where . Polynomials and rational functions constitute elementary, but important, classes of functions of a complex variable. n z zQzP )(/)( z zQ ≠ 0)( A generalization of the concept of function, called a multiple-valued function, is a rule that assigns more than one value to a point in the domain of definition. These multiple-valued functions occur in the theory of functions of a complex variable, just as they do in the case of real variables. When multiple-valued functions are studied, usually just one of the possible values assigned to each point is taken, in a systematic manner, and a single-valued function is constructed from the multi-valued function. z Example 4. Let denote any nonzero complex number. We know from Sec. 1.8 that has the two values: z 2/1 z
35Chapter IIAnalytic FunctionsHW= 21/2 = ±/rexpl2where r== and (-元<≤元) is the principal value of Argz,This formula gives amultiple-valued function,which is indeed two-valued.But, ifwe choose only the positive value of±rand write0W= f(2)= rexpl19(r>0,-元<0≤元),(2.1.3)then we obtain a single-valued function f defined on the set of nonzero numbers in thez -plane. Since zero is the only square root of zero, we also write f.(O)= O. The function fis then well defined on the entire plane. Similarly, we can get another single-valued function f.as follows.6w= f.()=-/rexpli9(r>0,一元<0≤元)(2.1.4)2
Chapter Ⅱ Analytic Functions 35 ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ ±== 2 exp 2/1 θ irzw , where = zr || and θ ( π θ ≤<− π ) is the principal value of . This formula gives a multiple-valued function, which is indeed two-valued. But, if we choose only the positive value of Argz ± r and write ,0( ) 2 exp)( πθπ θ ⎟ ≤<−> ⎠ ⎞ ⎜ ⎝ ⎛ + == rirzfw , (2.1.3) then we obtain a single-valued function defined on the set of nonzero numbers in the -plane. Since zero is the only square root of zero, we also write +f z = 0)0( +f . The function is then well defined on the entire plane. Similarly, we can get another single-valued function as follows. +f −f ,0( ) 2 exp)( πθπ θ ⎟ ≤<−> ⎠ ⎞ ⎜ ⎝ ⎛ − −== rirzfw . (2.1.4)
s2.2.Mappings1. Definition of a mappingOne can, however, display some information about the function by indicating pairs ofcorrespondingpoints ==(x,y)and w=(u,v).Todothis,it isgenerally simplertodrawthez-plane and w-plane,separately.See thefollowing figuref(D)f(z)0x0uz-planew-planeFig. 2-1isuouginurnsway,ntistellieiecuasamapping,orlld lucuunlJtransformation.The image ofa point = in the domain of definition D is the point w= f(-),and the set of images of all points in a set T contained in D is called the image of T anddenotedbyf(T).Thus,f(T)= (f(-): zeT)Theimagef(D)of theentiredomainof definition Dis justtherangeof fanddenotedbyran(f).The inverse imageofa point wis the setof all points zinthedomain of definitionof f that have w as their image. The inverse image of a point may contain just one point.many points, or none at all. The last case occurs, of course, when w is not in the range of fFora subset Bof the plane,the set of all inverse images of thepoints in B is called theinverse image of B, denoted by f-(B). Thus, f-'(B)=(ze D: f(-)e B).2.ExamplesExample 1. According to Example 2 in Sec. 2.1, the mapping w = -2 can be thought of asthetransformationu=x2-y2, v=2xy(2.2.1)from the xy-plane to the uv-plane.This form ofthe mapping is especially useful in finding theimagesofcertainhyperbolas.It is easy to show, for instance, that each branch of a hyperbolax2 - y2 =c (ci >0)(2.2.2)is mapped in a one to one manner onto the vertical line u = c.We start by noting from the firstof equations (2.2.1)that u=c, when (x,y) is a point lying on either branch.When, inparticular, it lies on the right-hand branch, the second of equations (2.2.1) tells us thatv= 2y /y’ +c, . Thus the image of the right-hand branch can be expressed parametrically asu=c,v=2yy2+c, (-00<y<0);and it is evident that the image of apoint (x,y)on thatbranchmoves upward along the entire
§2.2. Mappings 1. Definition of a mapping One can, however, display some information about the function by indicating pairs of corresponding points = yxz ),( and = vuw ),( . To do this, it is generally simpler to draw the z -plane and w -plane, separately. See the following figure. When a function is thought of in this way, it is often referred to as a mapping, or transformation. The image of a point in the domain of definition is the point , and the set of images of all points in a set f z D = zfw )( T contained in D is called the image of T and denoted by Tf )( . Thus, Fig. 2-1 = ∈TzzfTf }:)({)( . The image of the entire domain of definition is just the range of and denoted by . The inverse image of a point is the set of all points in the domain of definition of that have as their image. The inverse image of a point may contain just one point, many points, or none at all. The last case occurs, of course, when is not in the range of . For a subset Df )( D f f )ran( w z f w w f B of the plane, the set of all inverse images of the points in B is called the inverse image of B , denoted by )( . Thus, . 1 Bf − })(:{)(1 ∈∈= BzfDzBf − 2. Examples Example 1. According to Example 2 in Sec. 2.1, the mapping can be thought of as the transformation 2 = zw 2, xyvyxu (2.2.1) 22 =−= from the xy -plane to the -plane. This form of the mapping is especially useful in finding the images of certain hyperbolas. uv It is easy to show, for instance, that each branch of a hyperbola )0( (2.2.2) 11 22 ccyx >=− is mapped in a one to one manner onto the vertical line 1 = cu . We start by noting from the first of equations (2.2.1) that when is a point lying on either branch. When, in particular, it lies on the right-hand branch, the second of equations (2.2.1) tells us that 1 = cu yx ),( 1 2 2 += cyyv . Thus the image of the right-hand branch can be expressed parametrically as 2, ( ) 1 2 1 cyyvcu y ∞<<−∞+== ; and it is evident that the image of a point on that branch moves upward along the entire yx ),(
line as (x,y)traces out the branch in the upward direction (Fig.2-2).Likewise, since the pair ofequationsu=cCi, v=-2yy2 +c (-00<y<o0)furnishesaparametricrepresentation fortheimage of theleft-handbranchof thehyperbola,theimage of a point going downward along the entire left-hand branch is seen to move up the entirelineu=C,LVu=c,>0V=C,>0中u0Fig. 2-2On the other hand, each branch of a hyperbola2xy=C2 (C2 >0)(2.2.3)is transformed into the line v = C2, as indicated in Fig.2-1.We shallnowuseExample1tofind the imageofacertain regionExample 2. The domain D= ((x,y): x>0,y>0,xy<1) consists of all points lyingon the upper branches of hyperbolas from the family 2xy =C, where 0<c<2(Fig.18).WeknowfromExample1thatasapointtravelsdownward alongtheentiretyof oneofthesebranchesits image under the transformation w=zmoves to the right along the entire line = c.Since,forall valueofcbetween0and 2,thebranchesfill out thedomain((x,y):x>0,y>0,xy<1),that domain is mapped onto the horizontal strip ((u, v) : 0<v<2)In view of equations (2.2.1), the image of a point (0,y) in the z-plane is (-y2,0)Hence as (O,y) travels downward to the origin along the y axis, its image moves to the rightalong the negative u axis and reaches the origin in the w-plane.Then, since the image of apoint (x,O) is (x2,0), that image moves to the right from the origin along the u axis as(x,O) moves to the right from the origin along the x axis. The image of the upper branch of thehyperbola xy =1 is, of course, the horizontal line =2.Evidently, the closed regionD= ((x,y): x ≥0,y≥0,xy≤1)is mapped onto the closed strip D'= ((u, v): 0≤v≤2), as indicated in Fig. 2-3.VI山D2iE'D'ExB'CiuBCFig. 2-3
line as traces out the branch in the upward direction (Fig. 2-2). Likewise, since the pair of equations yx ),( 2, ( ) 1 2 1 cyyvcu y ∞<<−∞+−== furnishes a parametric representation for the image of the left-hand branch of the hyperbola, the image of a point going downward along the entire left-hand branch is seen to move up the entire line . 1 = cu Fig. 2-2 On the other hand, each branch of a hyperbola )0(2 = ccxy 22 > (2.2.3) is transformed into the line , as indicated in Fig. 2-1. 2 = cv We shall now use Example 1 to find the image of a certain region. Example 2. The domain = >> xyyxyxD < }1,0,0:),{( consists of all points lying on the upper branches of hyperbolas from the family 2 = cxy , where < c < 20 (Fig. 18). We know from Example 1 that as a point travels downward along the entirety of one of these branches, its image under the transformation moves to the right along the entire line . Since, for all value of between 0 and 2, the branches fill out the domain 2 = zw = cv c >> xyyxyx < }1,0,0:),{( , that domain is mapped onto the horizontal strip < vvu < }20:),{( . In view of equations (2.2.1), the image of a point in the -plane is . Hence as travels downward to the origin along the axis, its image moves to the right along the negative axis and reaches the origin in the -plane. Then, since the image of a point is , that image moves to the right from the origin along the u axis as moves to the right from the origin along the y),0( z )0,( 2 −y y),0( y u w x )0,( )0,( 2 x x )0,( x axis. The image of the upper branch of the hyperbola xy = 1 is, of course, the horizontal line v = 2 . Evidently, the closed region = ≥≥ xyyxyxD ≤ }1,0,0:),{( is mapped onto the closed strip ′ = ≤ vvuD ≤ }20:),{( , as indicated in Fig. 2-3. Fig. 2-3