Home_10 - Homework 10 Stephen Taylor June 9, 2005 Page 83...

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Unformatted text preview: Homework 10 Stephen Taylor June 9, 2005 Page 83 3. Let p ( z ) be a polynomial of degree n and let R > 0 be sufficiently large so that p never vanishes in { z : | z | R } . If ( t ) = Re it , 0 t 2 , show that Z p ( z ) p ( z ) dz = 2 in We represent the partial fractions decomposition of p /p as a 1 z- z 1 + + a n z- z n Since we may choose p to be monic, we multiply through by it to find p ( z ) = a 1 ( z- z 2 ) ( z- z n ) + + a n ( z- z 1 ) ( z- z n- 1 ) Equating leading coefficients we find that the sum of the a i is n . So distributing the line integrals over each term we find that the value of the desired integral is 2 i X a i = 2 in Page 87 1. Suppose f : G C is analytic and define : G G C by ( z,w ) = [ f ( z )- f ( w )]( z- w )- 1 if z 6 = w and ( z,z ) = f ( z ). Prove that is continuous and for each fixed w , z ( z,w ) is analytic....
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This note was uploaded on 10/19/2009 for the course MATH 814 taught by Professor Cong during the Three '09 term at University of Adelaide.

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Home_10 - Homework 10 Stephen Taylor June 9, 2005 Page 83...

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