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