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Unformatted text preview: ACM95a/100a Last updated October 14, 2009 Problem Set II Solutions Prepared by Faisal Amlani 1. (10 pts.) For a given real number , 0 < 2 , find the image of the sector 0 arg( z ) < under the transformation w = z 4 . How large should be so that the w plane is covered exactly once? SOLUTION : In polar coordinates, the transformation is given as w = z 4 = ( r e i ) 4 = r 4 e i 4 . So we see that w : { r e i  r , < } { r 4 e i 4  r , < } = { r e i  r , < 4 } . So the image of the sector 0 arg( z ) < is w : { z  arg( z ) < } { z  arg( z ) < 4 } If = / 2 , then { z  arg( z ) < 4 } { z  arg( z ) < 2 } and the map of the sector will cover exactly once. 2. (10 pts.) Let negationslash = 0, negationslash = 0 be two complex numbers. Show that = t for some real number t (i.e. the vectors defined by and are parallel) if and only if ( ) = 0. SOLUTION : : If = t , then = t   2 . This is a real number which implies ( ) = 0. : Assume that ( ) = 0. Then = r for some r R . Multiplying by gives   2 = r = = r   2 = = t for t = r   2 . 3. (25 pts.) Show that cos 1 ( z ) = i log ( z + i (1 z 2 ) 1 / 2 ) . Explain how to introduce cuts to give a branch with cos 1 (0) = / 2. SOLUTION : Cos 1 ( x ) is shown in Figure 1 for real variables in the range [ 1 , 1]....
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This note was uploaded on 01/14/2010 for the course ACM 1 taught by Professor Prof during the Fall '09 term at Caltech.
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