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380_solns_3[1] - Math 380 Solutions to Assignment 3 2.1.13...

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Math 380 Solutions to Assignment 3 2.1.13 Let w = z 2 : We°ll use the notation z = x + iy and w = u + iv: (a) The image of the line x = 1 is given by w = (1 + iy ) 2 = 1 ° y 2 + 2 yi where y ranges over all real values. These points in the uv plane are on the curve u = 1 ° v 2 = 4 , which is a parabola. For a more careful analysis, you can use parametric equations: The line x = 1 is parametrized by x = 1 and y = t; °1 < t < 1 : The image of this line is given by w = (1 + it ) 2 = 1 ° t 2 + 2 it which can be written parametrically as u = 1 ° t 2 and v = 2 t: Eliminating t , we get u = 1 ° v 2 = 4 , which is a parabolic curve in the uv -plane. 0 -5 -10 -15 -20 1 0 5 0 -5 -10 u v Note that as t varies over the interval ( °1 ; 1 ) ; the value of v ranges from °1 to 1 : This ±lls out the complete parabola moving upward at all times. (b) The hyperbola xy = 1 can be written as z = x + i x : So its image is given by w = ° x + i x ± 2 = x 2 ° 1 x 2 + 2 i . Thes values of w all lie on the line v = 2 : (c) The curve j z ° 1 j = 1 can be parametrized by z = 1 + e it ; 0 ± t ± 2 °: Its image is then given by w = ° 1 + e it ± 2 = 1 + 2 cos t + cos 2 t + i
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