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Unformatted text preview: 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.-5-10-15-20 10 5-5-10 u v 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 ; t...
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This note was uploaded on 09/16/2011 for the course MATH 380 taught by Professor Staff during the Spring '11 term at S.F. State.
- Spring '11