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Unformatted text preview: Key to Complex Analysis Homework 1 Chapter 1 (in Greene & Krantzs book) 13. Give all possible polar forms of each number. solution. (a) z = 3 + i : r =  z  = 3 + 1 = 2 = arg( z ) = arctan( = z < z ) + 2 k = 6 + 2 k, k Z Hence, all the possible polar form of z can be written as: z = re i = 2 e 6 i +2 ki , k Z . (d) z = 4 8 i : r =  z  = 16 + 64 = 4 5 , = arg( z ) = arctan( = z < z ) + 2 k = arctan(2) + 2 k, k Z Hence, all the possible polar form of z can be written as: z = re i = 4 5 e arctan(2) i +2 ki , k Z . 2 14. Convert the polar forms into rectangular forms. solution. (b) z = 4 e i/ 4 Since r = 4 , = / 4, we have that x = r cos( ) = 4 * 2 2 = 2 2 y = r sin( ) = 4 * 2 2 = 2 2 Hence, z = 2 2 + 2 2 i. (g) Since r = 7 , = / 6, we have that x = r cos( ) = 7 * 3 2 = 7 3 2 y = r sin( ) = 7 * 1 2 = 7 2 Hence, z = 7 3 2 7 2 i. 1 2 28. Compute each of the following derivatives: (a) z ( 4 z 2 z 3 ) (b) z ( z 2 + z 2 z 3 ) (c) 3 x 2 y ( 3 z 2 z 4 2 z 3 z + z 4 z 5 ) (d) 5 z 3 z 2 ( z 2 z z + 4 z 6 z 2 ) Solution. We use the fact that z z = 1 , z z = 0 , z z = 0 , z z = 1 , as well as the usual sum and product rule for differentiation....
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 Summer '08
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