MA530_AUG95

# MA530_AUG95 - f is an entire function such that for every...

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MATH 530 Qualifying Exam August 1995 Notation: D 1 (0) denotes the unit disk, { z C : | z | < 1 } . 1. A famous sequence of numbers is deﬁned by c 0 =0, c 1 =1,and c n = c n - 1 + c n - 2 for n =2 , 3 , 4 ... Prove that the c n are Taylor coeﬃcients at the origin of the rational function, z/ (1 - z - z 2 ). What is the radius of convergence of the series? 2. Find an analytic function that maps Ω = D 1 (0) - [0 , 1] one-to-one and onto the left half-plane H = { z C :R e z< 0 } . Is the mapping you found unique? Explain. 3. Suppose that f is a continuous function on { z C :Im z 0 } that is analytic on { z C :Im z> 0 } . Show that if f vanishes on a non-empty interval ( a,b )on the real axis, then f must vanish identically. Is the same result true if the word “analytic” is replaced by the word “harmonic?” Explain. 4. Evaluate Z 0 cos ax - cos bx x 2 dx where a and b are positive real constants. Hint: Integrate e iaz - e ibz z 2 around the contour below. (Prove any limits you use). C R C ε ε R . 0 5.
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Unformatted text preview: f is an entire function such that for every compact set K ⊂ C , the inverse image f-1 ( K ) is also compact. Prove that f ( C ) = C . 6. Suppose that f is a non-vanishing analytic function on the complex plane with the two points ± 1 deleted. Let γ 1 denote the curve given by z 1 ( t ) = 1+ e it where ≤ t ≤ 2 π and let γ 2 denote the curve given by z 2 ( t ) =-1+ e it where 0 ≤ t ≤ 2 π . Suppose that 1 2 πi Z γ j f ( z ) f ( z ) dz is divisible by 2 for j = 1 , 2. First, explain why 1 2 πi Z γ f ( z ) f ( z ) dz must be divisible by 2 for any closed curve in C-{± 1 } . Next, prove that f has an analytic square root on C-{± 1 } , i.e., show that there is an analytic function g on C-{± 1 } such that g 2 = f ....
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