math185f08-hw6

# math185f08-hw6 - f z 2 = f z 2 for all z ∈ C 4 If f is...

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MATH 185: COMPLEX ANALYSIS FALL 2008/09 PROBLEM SET 6 1. (a) Let a R . Show that lim r →∞ Z r - r e - ( x + ia ) 2 dx exists and is independent of a . (b) Let f be analytic on D (0 , 1). Suppose [Re f (0)] 2 = [Im f (0)] 2 . Show that Z 2 π 0 [Re f ( re )] 2 = Z 2 π 0 [Im f ( re )] 2 for 0 < r < 1. 2. Evaluate the following integrals: Z 2 π 0 e e dθ, Z 2 π 0 e ( e - ) dθ, Z 2 π 0 | 2 e - 1 | 2 , Z 2 π 0 | e - 2 | 4 . 3. (a) Let f ( z ) = n =0 a n z n and g ( z ) = n =0 b n z n be power series with positive radii of conver- gence r f and r g respectively. Deﬁne the product function f · g : Ω C by f · g ( z ) = f ( z ) g ( z ) for all z Ω := D (0 ,r f ) D (0 ,r g ). Prove that f · g has a power series representation f · g ( z ) = X n =0 c n z n , where c n = X n m =0 a m b n - m , with its radius of convergence is at least min { r f ,r g } . (b) Determine all entire functions f that satisfy
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Unformatted text preview: f ( z 2 ) = [ f ( z )] 2 for all z ∈ C . 4. 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. Date : November 1, 2008 (Version 1.0); due: November 7, 2008. 1...
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