By C. W. Celia, A. T. F. Nice, K. F. Elliott
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Additional info for Advanced Mathematics 3
10 When w = l /z and z = x 56 Advanced mathemat ics 3 = = 4 is mapped to the circle ~ l iz maps the circle [z - 31 + iy, w = u + iv, show that x 2. Given that = 21z1 to the = u/(u 2 + v2 ) . Find the area of the region in the w-plane to which the transformation w = l iz maps the region given by I ~ x ~ 2. 11 + b The transformation w = --~ cz + d az + b d ' where a, b, e, d E C, the set of complex numbers, and ez + z =I- - die . 9. If ale = bid, the relation between wand z reduces to w = ale for all values of z.
X, and the n roots of z" = I are I, 2re . 2re cos - ± I sm - , n n 4re .. 4re cos - ± Ism - , n n -1. In Fig. 3 the points A, B, C, D, E and F on the unit circle represent the roots of the equation Z6 = 1. These roots are 1, cos 60· ± i sin 60·, cos 120· ± i sin 120" Case 2 n odd The required values of k are 0, ± 1, ±2, The corresponding values of 0 are 30 Advanced mathematics 3 . , ±(n - 1)/2. -1. r D A x Roots of z6 = 1 Fig. , ± I)n (n - , n and the n roots are I, 2n cos n ± cos . 2n , n I Sin - 4n n COS - ± .
8 Show that the equation arg [: : ~J = ~ represents an arc of a circle of radius 4. Draw this arc in an Argand diagram, together with the half-lines arg (z - 2) = 2n/3 and arg (z + 2) = n/2. Complex numbers 47 9 The circle [z - 41 = 5 cuts the y-axis at the points A and B. Show that the equation of the major arc of this circle standing on AB is arg [z + 3~J = z - 31 tan -I (3/4). 10 Shade in an Argand diagram the region given by 1J z --. rr/3 ~ arg [ Z+l Show that the area of this region is (4rr ~ 2rr/3.