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Unformatted text preview: Math 185 Solutions to Midterm II 1. (a) The polynomial z 4 + 1 factors as a product ( z a 1 ) ( z a 4 ) where a 1 , . . . , a 4 are its four roots. These are easy to find in polar coordinates since, if z = re i ,we have z 4 = 1 iff r 4 e 4 i = e i so r = 1 and 4 = + 2 k for k Z . As a result the four roots are: e i 4 , e i 3 4 , e i 5 4 , e i 7 4 and the polynomial can be factored as z 4 + 1 = ( z e i 4 )( z e i 3 4 )( z e i 5 4 )( z e i 7 4 ) (b) The distance between any two of the above roots is at least 2 so the circles C 1 and C 2 contain only the roots e i 4 and e i 3 4 respectively. Now we will study the integrals of 1 z 4 +1 around C 1 and around C 2 independently. Define g ( z ) = 1 ( z e i 3 4 )( z e i 5 4 )( z e i 7 4 ) . This is an analytic function at all points where the denominator does not vanish so in particular it is analytic in C 1 and its interior. As a result, we can apply the Cauchy integral formula and obtain Z C 1 1 z 4 + 1 dz = Z C 1 g ( z ) ( z e i 4 ) dz = 2 ig ( e i 4 ) Similarly we can define h ( z ) = 1 ( z e i 4 )( z e i 5 4 )( z e i 7 4 ) and using the same reasoning as in the last bullet conclude that Z C 2 1 z 4 + 1 dz = Z C 2 h ( z )...
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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.
 Fall '07
 Lim
 Factors, Polar Coordinates

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