- 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0,-元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 ≤ ⎠ ⎞ ⎜ ⎝ ⎛ + == 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 (-000V=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,xy0,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 ∞ (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 > 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
Our lastexample illustrates how polar coordinates canbe used in analyzing certainmappingsExample 3.The mapping w==2becomes w=r ei2ewhen z=ree.Hence, if wewrite w = pe, then we have pei = r2er20; and Propositon 1.8.1(2) tlls us thatp=r2andΦ=20+2k元,where k has one of the values k =O,±1,+2,....Evidently, then, the image of any nonzeropoint zisfoundbysquringthemodulus of z and doublingavalueof Argz.Observe that points 2 = roe'° on a ciceler r = ro are transformed into points W= re/2on the circle p = r? As a point on the first circle moves counterclockwise from the positive realaxis to the positive imaginary axis, its image on the second circle moves counterclockwise fromthe positive real axis to the negative real axis (see Fig.20). So, as all possible positive values ofro are chosen, the corresponding arcs in the z-plane and w-plane fill out the first quadrant andthe upper half plane, respectively. The transformation w= z? is, then, a one to one mapping ofthe first quadrant (r,0):r≥0.0≤≤元/2)in the z-plane onto the upper-half plane((p,Φ):p≥0,0≤Φ≤) in the w-plane, as indicated in Fig.2-4. The point z=0 is, ofcourse, mapped onto the point w = O.yyi0=z?o0r?xrouFig. 2-4The transformation w=z?alsomaps theupper half plane((r,0):r≥0,0≤≤元) onto theentirew-plane.However, in this case, the transformation is not one to one since both the positiveand negativereal axes inthe z-plane are mapped onto thepositive real axis in thew-planeWhen n is a positive integer greater than 2, various mapping properties of the transformw= z", ie., pe = r"ete, are similar to those of w = 2?. Such a transformation maps theentire z-plane onto the entire w-plane, where each nonzero point in the w-plane is the imageof n distinct points in the z -plane. The circle r=ro is mapped onto the circle p=r; andthe sector ((r,0): 0≤r ≤ro,0≤≤2元 / n) is mapped onto the disk p≤ r", but not in aonetoonemanner
Our last example illustrates how polar coordinates can be used in analyzing certain mappings. Example 3. The mapping becomes when . Hence, if we write , then we have ; and Propositon 1.8.1(2) tells us that 2 = zw θ = i22 erw iθ = rez φ ρ= i ew φ θ ρ i i22 = ere 2 ρ = r and φ = θ + 22 kπ , where k has one of the values k = ± ± ,2,1,0 K. Evidently, then, the image of any nonzero point z is found by squring the modulus of z and doubling a value of Argz . Observe that points on a circler iθ erz = 0 0 = rr are transformed into points on the circle . As a point on the first circle moves counterclockwise from the positive real axis to the positive imaginary axis, its image on the second circle moves counterclockwise from the positive real axis to the negative real axis (see Fig. 20). So, as all possible positive values of are chosen, the corresponding arcs in the -plane and -plane fill out the first quadrant and the upper half plane, respectively. The transformation is, then, a one to one mapping of the first quadrant θ = 22 0 i erw 2 0 ρ = r 0r z w 2 = zw rr ≥θ ≤ θ ≤ π }2/0,0:),{( in the z -plane onto the upper-half plane ≤≥ρφρ φ π≤ }0,0:),{( in the -plane, as indicated in Fig.2-4. The point is, of course, mapped onto the point . w z = 0 w = 0 Fig. 2-4 The transformation also maps the upper half plane 2 = zw θ rr π≤θ≤≥ }0,0:),{( onto the entire -plane. However, in this case, the transformation is not one to one since both the positive and negative real axes in the -plane are mapped onto the positive real axis in the -plane. w z w When is a positive integer greater than 2, various mapping properties of the transform , i.e., , are similar to those of . Such a transformation maps the entire -plane onto the entire -plane, where each nonzero point in the -plane is the image of distinct points in the n n = zw φ θ =ρ inni ere 2 = zw z w w n z -plane. The circle 0 = rr is mapped onto the circle ; and the sector n r ρ = 0 }/20,0:),{( rrr 0 θ≤≤≤θ ≤ π n is mapped onto the disk , but not in a one to one manner. n r ρ ≤ 0
s2.3.TheExponentialFunctionanditsMappingProperties1.Operation of exponential functionThat chapter will start with the exponential functionei=e'e=e"(cosy+isiny)(z=x+iy)(2.3.1)the two factors e* and e being well defined at this time (see Sec.1.6). Note that, definition(2.3.1), which can also be writtereiy=eey,whereew=cosy+isinyis suggested by the familiar property e+2=e"e2ofthe exponential function in calculus2.ExamplesExample 1. The transformation w= e'" can be writtenpe = e'e', where z = x+iype'o.Thus, p=e*and Φ=y+2n,where n is some integer (see Sec.1.8), andandw=thistransformationcanbeexpressed intheform(2.3.2)p=e.=yThe image of a typical point z =(cr,y) on a vertical line x = c, has polar coordinatesp=expC, and @=y in the w-plane.That image moves counterclockwise around the circleshown in Fig.2-5 aszmoves up the line.The image of the line is evidently the entire circle,andeach point on the circle is the image of an infinite number of points, spaced 2元 units apart,alongthe line.1y=C20xAexp CiFig. 2-5A horizontal line y = C, is mapped in a one to one manner onto the ray = C2.To see thatthis is so, we note that the image of a point z=(x,c,)has polar coordinates p=e"and@ = C2. Evidently, then, as that point z moves along the entire line from left to right, its imagemoves outward along the entire ray = C2, as indicated in Fig. 2-5.Vertical and horizontal line segments are mapped onto portions of circles and rays,respectively,andimagesofvariousregionsarereadilyobtainedfromobservationsmadeinExample 1. This is illustrated in the following example.Example 2.Letus showthat thetransformation w=emaps the rectangular region((x,y):a≤x≤b,c≤y≤d)onto the region ((p,Φ): ea ≤p≤ eb,c≤≤ d). The two regions and corresponding parts oftheir boundaries are indicated in Fig. 2-6. The vertical line segment AD is mapped onto the arcp= ea, c≤≤d, which is labeled A'D'. The images of vertical line segments to the right ofAD and joining the horizontal parts of the boundary are larger arcs, eventually,the image of the
§2.3. The Exponential Function and its Mapping Properties 1. Operation of exponential function That chapter will start with the exponential function iyxzyiyeeee )()sin(cos (2.3.1) xiyxz +== += the two factors and being well defined at this time (see Sec.1.6). Note that, definition (2.3.1), which can also be written x e iy e iyxiyx = eee + , where yiyeiy += sincos is suggested by the familiar property of the exponential function in calculus. 21 21 xxxx = eee + 2. Examples Example 1. The transformation can be written , where z = ew iyxi =ρ eee φ z x += iy and . Thus, and φ ρ i = ew x ρ = e φ = + 2ny π , where is some integer (see Sec.1.8); and this transformation can be expressed in the form n ye . (2.3.2) x ,φρ == The image of a typical point ),( 1 = ycz on a vertical line 1 = cx has polar coordinates 1 ρ = exp c and φ = y in the -plane. That image moves counterclockwise around the circle shown in Fig. 2-5 as moves up the line. The image of the line is evidently the entire circle; and each point on the circle is the image of an infinite number of points, spaced w z 2π units apart, along the line. Fig. 2-5 A horizontal line is mapped in a one to one manner onto the ray 2 = cy 2 φ = c . To see that this is so, we note that the image of a point ),( 2 = cxz has polar coordinates and x ρ = e 2 φ = c . Evidently, then, as that point z moves along the entire line from left to right, its image moves outward along the entire ray 2 φ = c , as indicated in Fig. 2-5. Vertical and horizontal line segments are mapped onto portions of circles and rays, respectively, and images of various regions are readily obtained from observations made in Example 1. This is illustrated in the following example. Example 2. Let us show that the transformation maps the rectangular region z = ew ≤≤ ≤ ≤ dycbxayx },:),{( onto the region . The two regions and corresponding parts of their boundaries are indicated in Fig. 2-6. The vertical line segment is mapped onto the arc , :),{( dcee }, a b ≤φ≤≤ρ≤φρ AD a ρ = e φ ≤≤ dc , which is labeled ′DA ′ . The images of vertical line segments to the right of AD and joining the horizontal parts of the boundary are larger arcs; eventually, the image of the
line segment BC is the arc p = eb, c≤β≤ d, labeled B'C'. The mapping is one to one ifd-c<2元.Inparticular,ifc=0andd=元,then0≤@≤元,andtherectangularregionismappedontohalfofacircularringV-y=C200uxexpciFig. 2-6Ourfinal examplehereusesthe images ofhorizontal lines tofind the imageof a horizontalstrip.Example3.Whenw=e",the image of theinfinite strip 0≤y≤元istheupperhalfv ≥ 0 of the w -plane (Fig. 2-7). This is seen by recalling from Example 1 how a horizontal liney=c is transformed into a ray =c from theorigin.Asthereal number c increases fromc = 0 to c = π, and the angles of inclination of the rays increase from Φ = 0 to Φ = 元川元ict+uololFig. 2-7
line segment BC is the arc , b ρ = e ≤ φ ≤ dc , labeled ′CB ′ . The mapping is one to one if cd <− 2π . In particular, if c = 0 and d = π , then 0 ≤ φ ≤ π ; and the rectangular region is mapped onto half of a circular ring. Our final example here uses the images of horizontal lines to find the image of a horizontal strip. Fig. 2-5 Fig. 2-6 Example 3. When , the image of the infinite strip z = ew 0 ≤ y ≤ π is the upper half v ≥ 0 of the w -plane (Fig. 2-7). This is seen by recalling from Example 1 how a horizontal line y = c is transformed into a ray φ = c from the origin. As the real number increases from to c c = 0 c = π , and the angles of inclination of the rays increase from φ = 0 to φ = π . Fig. 2-7
$2.4. LimitsIn this section, we will discuss the limits of sequences and functions.1.Definition of a convergent sequenceLet (z.) be a sequence of complex numbers. If there exists a complex number z suchthat lim | =, -z - O, then we say that (=,} is convergent and call the number z to be thelimit ofthe sequence (,), written limz, =z,or z,→z(no0)Proposition 2.4.1. lim z, = z lim Rez,= Rez,lim Imz,=ImzProposition2.4.2.Let(=,),(w,)besequencesofcomplex numbers.(1) If (-n) is convergent, then it is bounded, i.e., there is constant M>O such thatIz, <M forall neN;(2)Jf limz,=z and limw,=w,thenlimcz, =cz(VceC), lim(z,±w,)=(z±w),and limz,w, =zw;2(3)Jf limzn=z and limW,=W+0,thenlimz,/w,=z/w2. Definition of limit of a functionDefinition 2.4.2. For a function f : D-→C and a point zo, if for each positive number,there isa positive number such that(2.4.1)zeD,0z-zk8=1f(z)-Wk8,then we say that wo is the limitof f(=) as z approaches zo and writelim f(=)=Wo, or f(z)→wo(z→z0)(2.4.2)Geometrically, this definition says that, for each -neighborhoodN(wo,8) = (w:/ w-woks)of wo,there is a deleted -neighborhoodN°(20,0)=(z:04z-z0k)of z such that f(DnN(=o,))cN(wo,), see, Fig.2-8.川lOx010Fig. 2-8Ineorem Z.4.1.Ij a lumut o a JunctionJdejined on D exists at a point zo,then it isuniqueExample 1. Let f(z) = iz /2, D = (z := k 1), then
§2.4. Limits In this section, we will discuss the limits of sequences and functions. 1. Definition of a convergent sequence Let be a sequence of complex numbers. If there exists a complex number such that , then we say that is convergent and call the number to be the limit of the sequence , written }{ n z z =− 0||lim∞→ zzn n }{ n z z }{ n z zzn n = ∞→ lim , or nzz →→ ∞)( n . Proposition 2.4.1. zz zzzz n n n n n n lim ⇔= = = ImImlim,ReRelim ∞→ ∞→ ∞→ . Proposition 2.4.2. Let }{},{ be sequences of complex numbers. wz nn (1) If }{ is convergent, then it is bounded, i.e., there is constant such that n z M > 0 Mzn || ≤ for all ∈ Nn ; (2) If zzn n = ∞→ lim and n ww n = ∞→ lim , then = ∈∀ C)(lim∞→ cczczn n , nn wzwz )()(limn ± = ± ∞→ , and zwwz ; nn n = ∞→ lim (3) If zzn n = ∞→lim and lim = ≠ 0 ∞→ n ww n , then wzwz nn n = //lim∞→ . 2. Definition of limit of a function Definition 2.4.2. For a function : Df → C and a point , if for each positive number 0 z ε , there is a positive number δ such that ∈ < − zzDz 0 ||0 , < δ ⇒ − |)(| < ε wzf 0 , (2.4.1) then we say that is the limit of as approaches and write w0 zf )( z 0 z 0 )(lim0 wzf zz = → , or )()( . (2.4.2) 0 0 →→ zzwzf Geometrically, this definition says that, for each ε -neighborhood }|:|{),( 0 0 ε = − wwwwN < ε of , there is a deleted w0 δ -neighborhood }||0:{),( 0 δ zzzzN 0 <−<= δ o of such that 0 z zNDf 0 δ )),(( ⊂ o I ),( 0 wN ε , see, Fig. 2-8. Theorem 2.4.1. If a limit of a function defined on exists at a point , then it is unique. f D 0 z Fig. 2-8 Example 1. Let = izzf 2/)( , = zzD < }1|:|{ , then
lim f(=) =(2.4.3)2→wheneverzDand0<z-1k2<82|Theorem 2.4.2. Let f be defined on D,then lim f(=)=wo if and only iflim f(z,)=wo whenever (z,)c DI(=o) with z, → zo(n-→)Exmple2If (2)==,then the limit lm() does notexistIndeed, when z, =(,0),=, =(O,}), we have2,=++10→0,z,=0+it→0asn→0.But lim f(=,)=1 and lim f(=)=-1. Thus, by Theorem 2.4.2, we know that the limitlim f(2)does not exist. See Fig. 2-10=-→0yAz"+x0ziFig. 2-10While definition (2.4.1) provides a means of testing whether a given point wo is a limit, itdoes not directly provide a method for determining that limit. Theorems on limits, presented in thenext section, will enable us to actually find many limits
2 )(lim1 i zf z = → . (2.4.3) <− ε 2 )( i zf whenever ∈ < zDz − < 2|1|0 and ε . Theorem 2.4.2. Let f be defined on D , then 0 )(lim0 wzf zz = → if and only if whenever with 0 )(lim wzf n n = ∞→ }{\}{ 0 zDzn ⊂ )( n 0 nzz →→ ∞ . Example 2. If z z zf )( = , then the limit )(lim does not exist. 0 zf z→ Indeed, when ),0(),0,( 1 1 n n n n ′ = zz ′′ = , we have 00,00 ′ n 1 n →+= ′ n ′ = + iziz 1 n → as n → ∞ . But ′ =1)(lim and ∞→ n n zf ′′ = − .1)(lim∞→ n n zf Thus, by Theorem 2.4.2, we know that the limit )(lim does not exist. See Fig. 2-10. 0 zf z→ Fig. 2-10 While definition (2.4.1) provides a means of testing whether a given point is a limit, it does not directly provide a method for determining that limit. Theorems on limits, presented in the next section, will enable us to actually find many limits. w0