By C. W. Celia, A. T. F. Nice, K. F. Elliott
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Extra resources for Advanced Mathematics 3
U r V C, C ° 8, 8 A x 0, z- plane Fig. 16 AI w-plane II w= 2z This transformation is a magnification by the scale factor k, with centre the origin. The shape and the orientation of any figure will be unchanged. Figure 2. I6 illustrates the case when w = 2z. 3. Let b = 0, so that w = az. Iwl = lal x [z] arg w = arg z + arg a. Then and The distance of any point in the z-plane from the origin will be multiplied by lal, and each point will be rotated through the angle arg a about the origin. In the transformation w = 2iz, Iwl = 21zl .
Since tan (5n/4) = tan (n/4) = I, each of the statements arg (z - 1) - arg (z - i) arg (z - I) - arg (z - i) and = = n/4 5n/4 · .. (1) · . (2) implies the statement (x - 1)2 + (y - 1)2 = I, · . (3) but neither statement (1) nor statement (2) is equivalent to statement (3). Example 6 Find the value of z at the point at which the circular arc 46 Advanced mathematics 3 arg ( zz -+ 2i) 3n . 2 = 4 mtersects the circle Izz -+ 2il 2 = } 2. z - 2i W=--. Let z +2 At the point of intersection, Iwl = } 2 and arg w = 3n/4.
Sketch these circles. 6 Express each of the following transformations as a magnification followed by a rotation and then a translation. (b) w = (4 + 4i)z + i, (a) w = -2i(z + i), + i)(z + i), (d) w = (1 + i)(3z - I + i). (c)w = the transformation w = 4z - 12i magnification followed by a translation, translation followed by a magnification. that the circle [z - 4il = I is mapped to the circle [w - 4il = 4. , Z Z =1= 0 When w = I/z, Iwl = I/izi. This shows that points on the circle [z] = I will be mapped to points on the circle Iwl = 1.
Advanced Mathematics 3 by C. W. Celia, A. T. F. Nice, K. F. Elliott