Use this example as a guide to help solve the problem, this Example 2 from Section 19 of complex variable Churchill textbook EXAMPLE 2. If /() = 2, then (4) Aw 2+AZ-2 3+ AZ - 2 AZ Az Az Az Az 58 ANALYTIC FUNCTIONS CHAP. 2 If the limit of Aw/Az exists, it can be found by letting the point Az - (Ax, Ay) approach the origin (0,0) in the Az plane in any manner. In par- ticular, as Az approaches (0,0) horizontally through the points (Ax. 0) on the real axis (Fig. 29). AZ = Ax +10 = AX - 10 = Ax + /0 = Az. In that case, expression (4) tells us that Aw AZ AL - 1. AZ Hence if the limit of Aw/As exists, its value must be unity. However, when Az approaches (0, 0) vertically through the points (0, Ay) on the imaginary axis, so that AZ = 0 +iAy =0-iAy = -(0+iAy) = -Az. we find from expression (4) that Aw Az Hence the limit must be -1 if it exists. Since limits are unique (Sec. 15), it follows that dw/dz does not exist anywhere. Ay (O. AY) (0. 0) (AX. 0) Ax FIGURE 29 Use the method in Example 2, Sec. 19, to show that f'(z) does not exist at any point z when (a) f(z) = Rez; (b) f (z) = Imz