Advanced Calculus fi..

602 Advanced Calculus, Fifth Editionb) Prove that, if w = f (z) is a linear fractional function and wl = f (zl), w2 = f (Q), "w3 = f (~31, w4 = f (~4). thenprovided at least three of zl, z2,z3, z4 are distinct. Thus the cross-ratio is invariantunder a linear fractional transformation.C) Let zl , z2, z3 be distinct and let WI, w2, wg be distinct. Prove that there is one and onlyone linear fractional transformation w = f (z) such that f (zl) = WI,f (z2) = w2.f (z3) = w3, namely the transformation defined by the equation[z. ZI, z2. 231 = [w, WI. w2, w31.d) Using the result of c) find linear fractional transformations taking each of the followingtriples of z values into the corresponding w's:i) z = 1 0 w = 0, i, 1;(ii) z = 0, 00, 1; w = w,O, i.10. Let C be a circle of radius a and center Q. Points P, P' are said to be -- inverses of eachother with respect to C if P is on the segment QP' or P' is on QP and QP . QP' = a2.Q and oo are also considered as a pair of inverse points. If C is a straight line, points P,P' symmetric with respect to C are called inverses of each other.a) Prove: P, P' are inverses with respect to C, if and only if every circle through P, P'cuts C at right angles.b) Prove: If P, P' are inverses with respect to C and a linear fractional transformationis applied, taking P to PI, P' to P;, C to CI,then PI, Pi are inverses with respectto CI . [Hint: Use a), noting that circles become circles and right angles become rightangles under a linear fractional transformation.]11. a) Let w = f (2) be a linear fractional transformation mapping 17.1 5 1 on Iwl5 1. Provethat f (z) has the form (8.98).[Hint: Let zo map on w = 0. By Problem 10 b) 1/20 mustmap on w = oo. The point z = 1 must map on a point w = eiB. Take zl = 1, z2 = zo,z3 = 1 /To in 9 c).]b) Prove that every transformation (8.98) maps lz 1 0 onto Im (w) 2 0 has theform (8.99) and that every transformation (8.99) maps Im ( z) 2 0 onto Im (w) 2 0.13. Prove that every linear fractional transformation of Im (z) 1.0 on I w 1 5 1 has form (8.100)and that every transformation (8.100) maps Im (z) 1. 0 onto I w 1 5 1.14. Let w = f (2) be analytic at zo and let f '(zo) = 0. . . . , f ("'(~0) = 0, f ("+l)(zo) # 0.Prove that angles between curves intersecting at zo are multiplied by n + 1 under the transformationw = f (z).[Hint: Near zo, w = wo + cn+1(t - zoy" + C~+Z(Z - +. ..,where cn+l # 0. A curve starting at zo at t = 0 with tangent vector (complex number)-+ 0a # 0 can be represented as z = zo + at + ct, 0 5 t 5 t1, where 6 = ~ (t)as t --+ O+. Show that the corresponding curve in the VJ plane can be representedas w = wo + ~,+~a"+lt"+' + cl(t)tn+'. Show that (w- wo)/tnfl has as limit, fort + 0+, a tangent vector to the curve at wo, and that this tangent vector has argumentarg cn+l +(n + 1) arg a. From this conclude that two curves in the z plane forming angle crat zo become two curves in the w plane forming angle (n + l)a at wo.]15. Prove the validity of test V. [Hint: Let c = a + ib. Let zl and zz = 2, + reia be distinctpoints of D. ThenC[ f (9)- f (zI)] = c 1:' f '(z) dz = eia ir~f '(L) dt,