Advanced Calculus fi..

4594 Advanced Calculus, Fifth EditionVI. Let f (2) = u + iv be analytic in D, and let D, be the half-plane: y > 0. If acomplex constant c and a real number a can be found, such thatRe [c(z- a) f '(z)] > 01 in D,, then f is one-to-one in DZ (see Problem 16 following Section 8.21).EXAMPLE 1 (Translations) The general form isw = z + a + bi(a, b real constants).Each point z is displaced through the vector a + bi, as in Fig. 8.24.EXAMPLE 2 (Rotation-Stretchings) The general form isw = A~;@z (A, a real constants, A > 0).If one writes z = re'e, w = pei@, so that p, c$then one hasare polar coordinates in the w plane,Thus distances from the origin are stretched in the ratio A to 1, while all figures arerotated about the origin through the angle a (Fig. 8.25).•Figure 8.24Translation.Figure 8.25Rotation-stretching.