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Home_10 - Homework 10 Stephen Taylor June 9 2005 Page 83 3...

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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 γ ( t ) = Re it , 0 t 2 π , show that γ p ( z ) p ( z ) dz = 2 πin 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 πi a i = 2 πin Page 87 1. Suppose f : G C is analytic and define φ : G × G C by φ ( z, w ) = [ f ( z ) - f ( w )]( z - w ) - 1 if z = w and φ ( z, z ) = f ( z ). Prove that φ is continuous and for each fixed w , z φ ( z, w ) is analytic. As we take the limit as z w , we find that φ f ( z ) by definition of the derivative, so function is indeed continuous.

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Home_10 - Homework 10 Stephen Taylor June 9 2005 Page 83 3...

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