Advanced Calculus fi..

622 Advanced Calculus, Fifth EditionOne can single out a particular mapping f by imposing additional conditions.For example, one can require that three specified points zl, ~ 2 ~ on 2 3 the real axiscorrespond to three specified points wl, 202, wg on the boundary of D, providedthe cyclic order of the triples matches the corresponding positive directions on theboundaries. The justification for this rule is given by the fact that one can mapthe half-plane onto itself to take three given points on the real axis to three givenpoints on the real axis, provided the cyclic orders match (cf. Problem 9 followingSection 8.21).The points z,, . . . , z,, oo play special roles in the transformations (8.1 15),(8.117), (8.118). In seeking a transformation of one of these types onto a givendomain, one can assume three of the points at convenient positions, provided thecyclic order chosen agrees with the sense-preserving property of the transformation.It will be found most convenient to always make oo a special point, so that h,+l # h 1and, for (8.1 18), h,,, # h, + 2n. The other two special points can be chosen, forexample, as 0 and 1.Maps onto circle with teeth. As a final example of a class of explicit one-to-oneconformal mappings, we mention functions w = H(z) having the following form:Here the numbers k, and a, are real and k, > 0,O < a, < 1 for s = 1, . . . , n; thenumbers z, , . . . , z, represent distinct points on the circle lzl = 1 ; the principalvalue of the a, power is used. The function H(z) is then analytic for lzl < 1 andis moreover one-to-one in this domain (Problem 6 below). The image domain isapproximately the region I w 1 < 1 plus n sharp "teeth" projecting from it; the pointsof the teeth are the images of the points zl , . . . , z,. The smaller each a,, the sharperthe corresponding tooth. Further properties of these functions and other mappingclasses are described in a paper by P. Erdos, F. Herzog, and G. Piranian, PacificCoast Journal of Mathematics, Vol. 1 (1951), pp. 75-82.PROBLEMS1. Verify that the following functions define one-to-one conformal mappings of the upperhalf-plane and determine the image domains:a) w = 2Log(z + 1) - Logzb) w =Log+ + 2 ~ o g ~ + 3 L -2Log(z-3) o g ~e) w = Jw (pficipal value).2. Determine a one-to-one conformal transformation of the half-plane Im (z) > 0 onto eachof the following domains:a) the domain bounded by the lines v = 0, v = 2 and the ray v = 1,0 ( u < co. [Hint:Seek a transformation of form (8.1 16), with z1 = 0, z2 = 1, hl = 0, h2 = 1, h3 = 2.1b) the domain bounded by the lines v = 0, v = 2 and the rays: v = 1, -co -= u ( - 1;v = 1, 1 5 u < co. [Hint: Use (8.116) with zl = -1, z2 = 0,z3 = p > 0, hl = 0,h2 = 1, h3 = 2, h4 = 1 and determine p so that the mapping is correct.]