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Unformatted text preview: 3. Suppose there is R > 1 so that h ( z ) is analytic in the disk { z :  z  < R } . Prove that if  h ( z )  1 for  z  1 and h (0) = 0 and h (1) = 1, then  h (1)  1. (Hint: you may wish to consider lim r 1( h (1)h ( r )) / (1r ).) 4. (a) Express the arctangent function in terms of the logarithm. (b) Let A ( z ) be the branch of the arctangent function that is analytic except for { z : Re( z ) = 0 ,  Im( z )  1 } and that has A (0) = . Find, justifying your work, lim t + Re( A ( t + i )) (where, as usual, t + means t is positive and real as it approaches 0). 5. Use the residue theorem to evaluate Z t 4 + t 4 dt Justify your answer by careful statements of your contours and estimates....
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This document was uploaded on 01/25/2012.
 Spring '09
 Math

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