math185f09-hw6

math185f09-hw6 - Z ∂D a,r e z z(1-z 3 dz(a if 0 ∈ D a,r...

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MATH 185: COMPLEX ANALYSIS FALL 2009/10 PROBLEM SET 6 1. Prove the Fundamental Theorem of Algebra using Cauchy’s theorem as follows. Suppose we have a polynomial with complex coefficients p ( z ) = n k =0 a k z k that has no zeros in C . (a) Consider the polynomial q ( z ) = n k =0 a k z k whose coefficients are conjugates of the corre- sponding coefficients of p ( z ). Show that f ( z ) = 1 p ( z ) q ( z ) is an entire function. (b) Let R > 0. Apply Cauchy’s theorem to the line integral along the closed curve L in Figure 1b to show that Z R - R dx | p ( x ) | 2 + i Z π 0 Re p ( Re ) q ( Re ) = 0 . Figure 1. L is a closed curve in the counter clockwise direction along the semicircle { Re it | 0 t π } and the line segment { z | - R Re z R, Im z = 0 } . (c) Show that as R → ∞ , we get Z -∞ dx | p ( x ) | 2 = 0 . (d) Why is this a contradiction? 2. Let r > 1 and let f be analytic in D (0 ,r ). Starting from the integral Z Γ ± 2 ± ² z + 1 z ³´ f ( z ) z dz, where Γ is the curve z : [0 , 2 π ] C , z ( t ) = e it , prove that 2 π Z 2 π 0 f ( e it )cos 2 ² t 2 ³ dt = 2 f (0) + f 0 (0) , 2 π Z 2 π 0 f ( e it )sin 2 ² t 2 ³ dt = 2 f (0) - f 0 (0) . Date : November 1, 2009 (Version 1.0); due: November 6, 2009. 1
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3. Let a C and r > 0. Evaluate the integral 1 2 πi
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Unformatted text preview: Z ∂D ( a,r ) e z z (1-z ) 3 dz (a) if 0 ∈ D ( a,r ) and 1 / ∈ D ( a,r ); (b) if 0 / ∈ D ( a,r ) and 1 ∈ D ( a,r ); (c) if 0 ∈ D ( a,r ) and 1 ∈ D ( a,r ). 4. (a) Let f and g be entire functions that satisfy | f ( z ) | < | g ( z ) | for all z ∈ C . Show that there exists a constant λ ∈ C such that f ( z ) = λg ( z ) for all z ∈ C . (b) Determine all entire functions f that satisfies | f ( z ) | < | f ( z ) | for all z ∈ C . 5. If f is analytic on D (0 , 1) and let the power series expansion of f be f ( z ) = X ∞ n =0 a n z n . e below denotes the base of natural logarithms, ie. e = exp(1). (a) Suppose for all z ∈ D (0 , 1), | f ( z ) | ≤ 1 1- | z | . Show that for all n ∈ N , | a n | < ( n + 1) e. (b) Suppose for all z ∈ D (0 , 1), | f ( z ) | ≤ 1 1- | z | . Show that for all n ∈ N , | a n | < e. 2...
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This note was uploaded on 02/17/2010 for the course MATH 185 taught by Professor Lim during the Fall '07 term at Berkeley.

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math185f09-hw6 - Z ∂D a,r e z z(1-z 3 dz(a if 0 ∈ D a,r...

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