Advanced Calculus fi..

Chapter 8 Functions of a Complex Variable 603where z = zl + teIu on the straight-line path of integration. Show that the real partof the last integral is positive, so that f (z2) # f (zl) Cf. F. Herzog and G. Piranian,Proceedings of the American Mathematical Society, Vol. 2 (1951), pp. 625-633.116. Prove the validity of test VI. [Hint: Set zl = Log (z - a) and show that the functionwl = f [z(zl)] = g(z1) is analytic and satisfies the inequality: Re [cgl(zl)] > 0 in thedomain: 0 < Im (zl) c a. Now apply test V.]The following problem, known as the Dirichlet problem, arises in a variety of situationsin fluid dynamics, electric field theory, heat conduction, and elasticity: given adomain D, to$nd afunction u(x, y) harmonic in D and having given values on theboundary of D.The statement is somewhat loose as to the boundary values. It will be seenimmediately how it can be made more precise.Let D be a simply connected domain bounded by a simple closed curve C, as inFig. 8.32. Then assignment of boundary values is achieved by giving a function h(z)for z on C and requiring that u = h on C. If h is continuous, the natural formulationof the problem is to require that u(x, y) be harmonic in D, continuous in D plus C,and equal to h on C. If h is piecewise continuous, it is natural to require that u beharmonic in D, continuous in D plus C, except where h is discontinuous, and equalto h except at the points of discontinuity.If C is a circle, lzl = R, the problem is solved by the Poisson integral formulaof Section 5.17. For example, if R = 1, h can be written as h(8), a function of period.21r, anddefines a function harmonic in lzI < 1. If h(8) is continuous and we define u(l,8) tobe h(8), then u can be shown to be continuous for (z( 5 1 and therefore satisfies allcbnditions. If h(0) is discontinuous, the same procedure is successful and (8.101)again provides a solution.While the formula (8.101) does provide a solution of the problem, we must alsoask: Is this the only solution? The answer is yes when h is continuous for lzl = 1 ;the answer is no if h has discontinuities. In the latter case (8.101) provides a solutionFigure 8.32 Dirichlet problem, .