Advanced Calculus fi..

604 Advanced Calculus, Fifth EditionIu(x, y) which is bounded for (z < 1, and it can be shown that this is the only boundedsolution. With the supplementary requirement that the solution be bounded, (8.101)provides the one and only solution.Now let D be a domain bounded by C as in Fig. 8.32 and let us suppose that aone-to-one conformal mapping: zl = f (z) of D onto the circular domain lz I < 1 hasbeen found and that this mapping is also continuous and one-to-one on D plus C,taking C to (2, I = 1. By assigning to each point zl on lzl I = 1 the value of h at thecorresponding point on C, we obtain boundary values h 1 (7-1) on Jz 1 ( = 1. Let u(z1)solve the Dirichlet problem for lzl 1 < 1 with these boundary values. Then u[ f (z)]is harmonic in D and solves the given Dirichlet problem in D. For ~(zI) can bewritten as Re [F(z,)] where F is analytic and u [ f (z)] = Re (F[ f (z)]); i.e., u [ f (z)]is the real part of an analytic function in D. Hence u is harmonic in D. Sincecontinuous functions of continuous functions are continuous, u[ f (z)] will have theproper behavior on the boundary C and hence does solve the problem.Accordingly, conformal mapping appears as a powerful tool for solution of theDirichlet problem. For any domain which can be mapped conformally and one-tooneon the circle lzl < 1, the problem is explicitly solved by (8.101). It can be shownthat every simply connected domain D can be mapped in a one-to-one conformalmanner on the circular domain: 121 < 1, provided D does not consist of the entirez-plane. Furthermore, if D is bounded by a simple closed curve C, the mappingcan always be dejned on C so as to remain continuous and one-to-one. Proofs ofthese theorems appear in the following texts listed at the end of the chapter: Ahlfors(pp. 229-232), Bieberbach (1967, pp. 67-71), and Heins (pp. 130-134). For proofsof the stated properties of the Poisson integral formula, see Ahlfors (pp. 166-168)and Nevanlinna (pp. 22-24).There now remains the question of how to map a particular simply connecteddomain D onto a circular domain. Further information on this is given in Section8.27 below. We shall not consider extension of the theory to multiply connecteddomains.I8.23 DIRICHLET PROBLEM FOR THE HALF-PLANESince the transformationZI -iiz = - (8.102)ZI + imaps the domain lzl 1 < 1 on the half-plane Im (z) > 0, the Dirichlet problem for thehalf-plane is reducible to that for the circle as above. However, it is simpler to treatthe half-plane by itself. We shall develop the equivalent of (8.101) for the half-plane.Accordingly, if a domain D can be mapped on the half-plane, the Dirichlet problemfor D will be immediately solved.We consider first several examples.EXAMPLE 1 u(x, y) harmonic for y > 0; u = rr for y = 0, x < 0; u = 0 fory = 0, x > 0 as shown in Fig. 8.33. The function