M1sols - Math 185 Solutions to Midterm I 1. z 6- 1 2 + i 1...

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Unformatted text preview: Math 185 Solutions to Midterm I 1. z 6- 1 2 + i 1 2 = 0 if and only if z 6 = 1 2- i 1 2 . So we want to find all sixth roots of the number 1 2- i 1 2 = e- 4 . If z = re i , the equation z 6 = e- 4 becomes r 6 e i 6 = e- 4 so r = 1 and 6 =- 4 +2 k for some k Z . Thus the solutions to the equation are the complex numbers of the form e i (- 24 + 2 k 6 ) for k = 0 , 1 , . . . , 5. 2. (a) The function z (1+ i ) z +1 is a composition of the map w = (1+ i ) z (tilting by 4 +scaling by a factor of 2) and the translation w w +1. Hence it maps rectangles to rectangles. The image of S is thus the rectangle with vertices 1 , 2+ i, 2- 2 + i (2 +1) , 1- 2 + i 2 which are the images of the vertices 0, 1,1 + 2 i and 2 i respectively. (b) Since e z = e x cos ( y ) + ie x sin ( y ) The horizontal line segments y = 0 and y = 2 are both mapped to the ray 1 r e , = 0 and the vertical lines segments x = 0 and x = 1 are mapped to circles centered at the origin of radii 1 and e respectively. The region is mapped to the annulus 1 r e and 0 2 ....
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This note was uploaded on 09/01/2010 for the course MATH 185 taught by Professor Lim during the Fall '07 term at University of California, Berkeley.

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M1sols - Math 185 Solutions to Midterm I 1. z 6- 1 2 + i 1...

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