Theorem 8.2.2. If a finction f is conformal at a point zo, then it is unilateral on someneighborhood of zo.Proof. Since f'(z)+0, using Lemma 8.2.1 for m=1, we know that there exists aneighborhood N(zo,e) of z and a neighborhood N(wo,) of wo such thatza is theuniquezero off()-woinN(zo,)and it is of order 1,and for eachfixedpointweN(wo,), the function f(z)-w has exactly 1 zero in the deleted neighborhoodN(zo,). From the continuity of f(z)-w, we can choose a positive number μ< suchthatf(N(zo,μ) N(wo,8).If f is not unilateral on N(zo,μ), then there are two distinct points zj,=, E N(zo,μ) suchthat f(-)=f(=,)=w.Thus,z1,=2±zo,w±wo sincezo is the unique zero off(z)-wo in N(zo,).We see that the function f(z)-w has two distinct zerosZi,Z2 e N°(z0, μ)c N°(z0,6)Since we N(wo,), this is impossible.Thus, f is unilateral on N(zo,).The proof iscompleted.Theorem 8.2.3. Let f be a nonconstant analytic function on a domain D, then f(D)is a domain.Proof. Let wE f(D), then z e D such that wo = f(zo). Since D is open, there aneighborhood N(zo,R)D.Since is analytic and nonconstant in D,there is a positiveinteger m suchthatf'(=o)= "(z0)=..= f(m-1(=0) = 0, f(m)(z0)+ 0 .From Lemma 8.2.1, there is a positive number r < R such that V0<<r,38 >0 so that foreach fixed point weN(wo,),the function f(z)-whas exactly m zeros countingmultiplicity in the deleted neighborhood N°(zo,). Thus,N(wo,8)c f(N(z0,8) cDHence, wo is an interior pointof f(D) and so f(D) is open.For points w,w, E f(D), take z),=2 ED with (z)=w(k=1,2). Since D isconnected, there is a polygonal lineL: z= z(t)(a≤t≤b)contained in D such that z(a)=zi,z(b)=z2. Since f has conitinuous derivative in D,wegetapiecewise smooth curver: w= w(t)= f(z(t)(a≤t<b)contained in f(D) such thatw(a) = w,w(b)= W2The compactness of the I enable us to find a polygonal line 'contained in f(D) andjioning w, and w,:This proves that f(D) is a domain in the w-plane.The proof iscompleted
Theorem 8.2.2. If a function is conformal at a point , then it is unilateral on some neighborhood of . f 0 z 0 z Proof. Since 0)(′ zf 0 ≠ , using Lemma 8.2.1 for m =1, we know that there exists a neighborhood ),( 0 zN ε of and a neighborhood 0 z ),(wN 0 δ of such that is the unique zero of in w0 0 z 0 )( − wzf ),( 0 zN ε and it is of order 1, and for each fixed point ),( , the function wNw 0 δ o ∈ )( − wzf has exactly 1 zero in the deleted neighborhood ),( . From the continuity of 0 zN ε o )( − wzf , we can choose a positive number μ < ε such that ),()),(( 0 μ ⊂ wNzNf 0 δ . If f is not unilateral on ),(zN 0 μ , then there are two distinct points ),(, 21 ∈ zNzz 0 μ such that :)()( == wzfzf . Thus, 1 2 021 0 ≠ , ≠ wwzzz since is the unique zero of in 0 z 0 )( − wzf ),( 0 zN ε . We see that the function )( − wzf has two distinct zeros ),(),(, 21 0 0 μ zNzNzz ε o o ∈ ⊂ . Since ),( , this is impossible. Thus, is unilateral on wNw 0 δ o ∈ f ),(zN 0 μ . The proof is completed. Theorem 8.2.3. Let be a nonconstant analytic function on a domain , then is a domain. f D Df )( Proof. Let )( 0 ∈ Dfw , then ∃ ∈ Dz0 such that )( 0 0 = zfw . Since is open, there a neighborhood . Since is analytic and nonconstant in , there is a positive integer such that D 0 ),( ⊂ DRzN f D m )()( 0)(,0)( 0 )( 0 )1( ′ 0 = ′′ 0 == = ≠ − zfzfzfzf L m m . From Lemma 8.2.1, there is a positive number r < R such that ∀ < ε < r δ >∃ 0,0 so that for each fixed point ),( , the function wNw 0 δ o ∈ )( − wzf has exactly zeros counting multiplicity in the deleted neighborhood . Thus, m ),( 0 zN ε o 0 δ ⊂ 0 ε )),((),( ⊂ DzNfwN . Hence, is an interior point of and so is open. w0 Df )( Df )( For points )(, , take 21 ∈ Dfww ∈ Dzz 21, with = kwzf = )2,1()( kk . Since is connected, there is a polygonal line D = ≤ ≤ btatzzL ))((: contained in D such that 1 2 = )(,)( = zbzzaz . Since has conitinuous derivative in , we get a piecewise smooth curve f D Γ = = ≤ ≤ btatzftww )))((()(: contained in Df )( such that 1 2 = )(,)( = wbwwaw . The compactness of the Γ enable us to find a polygonal line Γ′ contained in and jioning and . This proves that is a domain in the -plane. The proof is completed. Df )( w1 w2 Df )( w
s8.3.Local InversesA transformation w= f(-) that is conformal at a point zo has a local inverse there. That is,we haveTheorem 8.3.1. If a transformation w= f() is conformal at a zo and w= f(z),then there exists a unique transformation z=g(w),which is defined and analytic in aneighborhood N of wo, such that g(wo)==o and flg(w))=w for all pints w in Nand the derivative of g(w) is, moreover,1g'(w) =(8.3.1)f(2)Proof. As noted in Sec. 8.1, the conformality of the transformation W=f(=) at zoimplies that there is a neighborhood N(zo,) of zo throughout which f is analytic andf'(-)+0.Henceifwewritez=x+iy, zo=xo +iyo,and f(z)=u(x,y)+iv(x,y),we know that there is a neighborhood N(zo,) of the point (xo,yo) throughout which thefunctions u(x,y) and v(x,y) along with their partial derivatives of all orders, are continuous(see Sec. 4.13).Now, the pair ofequationsu=u(x,y), v=v(x,y)(8.3.2)represents a transformation from the neighborhood N(zo,) into the uv plane. Moreover, theJacobiandeterminantluxu,J==uy, -vu, =(u) +(v) =f()Pyxyyis nonzero on N(=o,).Ifwewrite(8.3.3)o = u(xo,yo) and Vo = v(xo,yo)then from the theory of implicit functions in mathematical analysis, we know that there is a uniquecontinuous transformationx=x(u,v), y= y(u,v),(8.3.4)defined on a neighborhood N of the point (uo,vo) such thatu= u(x(u,v),y(u,v), = v(x(u,v),y(u,v), xo = x(uo,vo), yo = y(uo,vo)Also, the functions (8.3.4) have continuous first-order partial derivatives satisfying the equations111(8.3.5)xu=Vy,x,=uysyu=Vx,y=-uJ1Jthroughout N.Ifwewrite w=u+iv and wo=uo+ivo,aswell as(8.3.6)g(w)= x(u, v)+iy(u,v),the transformation z = g(w) is evidently the local inverse of the original transformationw= f(z) at zo.Transformations (8.3.2)and (8.3.4) can be writtenu+iv=u(x,y)+iv(x,y)andx+iy=x(u,v)+iy(u,v);and these last two equations are the same asw= f(=) and z= g(w),where g has the desired properties. From expression (8.3.5), we see that the function (8.3.4)have continuous partial derivatives and that the Cauchy-Riemann equationsx =y,x, =-yu
§8.3. Local Inverses A transformation = zfw )( that is conformal at a point has a local inverse there. That is, we have 0 z Theorem 8.3.1. If a transformation = zfw )( is conformal at a and , then there exists a unique transformation 0 z )( 0 0 = zfw = wgz )( , which is defined and analytic in a neighborhood N of , such that w0 00 )( = zwg and )]([ = wwgf for all pints in and the derivative of is, moreover, w N wg )( )( 1 )( zf wg ′ ′ = . (8.3.1) Proof. As noted in Sec. 8.1, the conformality of the transformation at implies that there is a neighborhood = zfw )( 0 z ),(zN 0 δ of throughout which is analytic and . Hence if we write 0 z f ′ zf ≠ 0)( z = x + iy , 000 = + iyxz , and = + yxivyxuzf ),(),()( , we know that there is a neighborhood ),(zN 0 δ of the point throughout which the functions and along with their partial derivatives of all orders, are continuous (see Sec. 4.13). ),( 00 yx yxu ),( yxv ),( Now, the pair of equations = yxuu ),( , = yxvv ),( (8.3.2) represents a transformation from the neighborhood ),(zN 0 δ into the plane. Moreover, the Jacobian determinant uv yxyx yx yx uvvu vv uu J −== 2 2 2 zfvu )()()( x x =+= ′ is nonzero on ),(zN 0 δ . If we write ),( 0 00 = yxuu and ),( 0 00 = yxvv . (8.3.3) then from the theory of implicit functions in mathematical analysis, we know that there is a unique continuous transformation = vuxx ),( , = vuyy ),( , (8.3.4) defined on a neighborhood N of the point ),( such that 00 vu = vuyvuxuu )),(),(( , = vuyvuxvv )),(),(( , ),( 0 00 = vuxx , ),( . 0 00 = vuyy Also, the functions (8.3.4) have continuous first-order partial derivatives satisfying the equations u y v J x 1 = , v y u J x 1 −= , u x v J y 1 −= , v x u J y 1 = (8.3.5) throughout N . If we write = + ivuw and 000 = + ivuw , as well as = + vuiyvuxwg ),(),()( , (8.3.6) the transformation is evidently the local inverse of the original transformation at . Transformations (8.3.2) and (8.3.4) can be written = wgz )( = zfw )( 0 z +=+ yxivyxuivu ),(),( and + = + vuiyvuxiyx ),(),( ; and these last two equations are the same as = zfw )( and = wgz )( , where has the desired properties. From expression (8.3.5), we see that the function (8.3.4) have continuous partial derivatives and that the Cauchy-Riemann equations g uvvu = , = −yxyx
are satisfied in N.Thus, g is analytic in N.Furthermore, we compute when w= f(z),11-(y,-iv)=_(u-iv)=g(w)=x, +iyu=ux+ivf(=)The proof is completed.Definition 8.3.1. The function g in Theorem 8.3.1 is called the local inverse of thefunction f at oClearly, unilateral function f on a domain D has an inverse function f-l: f(D)→Dthat equals to the local inverse of at every point zo.Thus, from Theorems 8.2.1, 8.2.3 and8.3.1, we haveCorollary 8.3.1. Every unilateral function f on a domain D has a unilateral inversef-1 on thedomain f(D) and1f-l(w)=VW= f(=)E f(D)f'()Corollary 8.3.2.Every unilateral function f on a domain D is a homoemorphism fromD onto f(D), i.e. f : D-→ f(D),f-l: f(D)-→D are continuous.Example 1. We saw in Example 1, Sec. 9.1, that if f()= e", the mapping w = f(2) isconformal everywhere in the z plane and, in particular, at the point zo =2πi. The image ofthis choice of zo is the point w=1.When points in the w plane are expressed in the formw=pexp(i),the local inverse at zo can be obtained bywriting g(w)=logw,wherelogwdenotesthebranchlogw=Inp+ig(p>0,元<0<3元)of the logarithmic function, restricted to any neighborhood of wo that does not contain the origin.Observe that g(1)=ln1+i2元 =2元i and that, when w is in the neighborhood,f[g(w)) =exp(logw) = wAlso,1dg(w) :0EFdwexpzWinaccordancewithequation(8.3.1)Note that, ifthe point zo=O is chosen, one can use the principal branchlogw=lnp+id(p>0,-元<0<元）of the logarithmic function to define g . In this case, g(l) = 0
are satisfied in N . Thus, g is analytic in N . Furthermore, we compute when = zfw )( , )( 11 )( 1 )( 1 )( zfivu ivu J ivv J iyxwg xx uu xy xx ′ = + ′ =−=−=+= . The proof is completed. Definition 8.3.1. The function g in Theorem 8.3.1 is called the local inverse of the function f at . 0 z Clearly, unilateral function on a domain has an inverse function that equals to the local inverse of at every point . Thus, from Theorems 8.2.1, 8.2.3 and 8.3.1, we have f D → DDff − )(: 1 f 0 z Corollary 8.3.1. Every unilateral function on a domain has a unilateral inverse on the domain and f D −1 f Df )( )()(, )( 1 )(1 Dfzfw zf wf ∈=∀ ′ = − . Corollary 8.3.2. Every unilateral function on a domain is a homoemorphism from onto , i.e. are continuous. f D D Df )( → → DDffDfDf − )(:),(: 1 Example 1. We saw in Example 1, Sec. 9.1, that if , the mapping is conformal everywhere in the plane and, in particular, at the point z )( = ezf = zfw )( z 2π iz0 = . The image of this choice of is the point . When points in the plane are expressed in the form 0 z w0 = 1 w = ρ iw φ)exp( , the local inverse at can be obtained by writing , where denotes the branch 0 z = log)( wwg log w = lnlog ρ + iw φ ρ > π < θ < π )3,0( of the logarithmic function, restricted to any neighborhood of that does not contain the origin. Observe that w0 g =π+= 221ln)1( πii and that, when w is in the neighborhood, wgf = )logexp()]([ = ww . Also, zw w dw d wg exp 11 ′ log)( === , in accordance with equation (8.3.1). Note that, if the point 0 is chosen, one can use the principal branch z0 = = lnlog ρ + iw φ ρ > ,0( −π < θ < π ) of the logarithmic function to define g . In this case, g = 0)1(