Advanced Calculus fi..

Advanced Calculus, Fifth EditionAs extreme cases one can choose m = 1 or m = n + 1. Let points xl, . . . , x, bechosen on the real axis such that xl < x2 < . . . < x,. A bounded harmonic functionv in the half-plane y > 0, with the following boundary values for y = 0:v =hlforx x,,here zl = xl + Oi, 2.2in the upper half-plane is= x2 + Oi, . . . . A corresponding f (z) = u + iv = w analytic8 'Nl+ hn+l {in- Log (z- zn)] + a,1where a is a real constant. When the constants hl, ..., h,+l satisfy (8.115), thisfunction gives a one-to-one conformal mapping of the half-plane; for a proof, seeProblem 5 below. The image domain Dl is bounded by lines v = const.EXAMPLE 1zLog(z+1)+3~~~-2+1+ I2(in - Log Z)maps the upper half-plane on the domain Dl shown in Fig. 8.43.I *U- 1 1Figure 8.43Range of example.