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MA530_AUG01 - r such that the estimate | φ z |< Me | z...

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MATH 530 Qualifying Exam August 2001 1. (20 pts) Suppose that f ( z ) is analytic in the unit disc D 1 (0) with f (0) = 0 and | f ( z ) | < 1 on D 1 (0). Prove that n =1 f ( z n ) converges to a function which is also analytic in D 1 (0). Be sure to carefully explain every step in your proof and to write out the statement of any theorems you refer to. 2. (20 pts) Evaluate the integral Z 0 ln x ( x + 2) 2 dx . Hint: Define a branch of log z and integrate (log z ) 2 ( z + 2) 2 around an appropriately chosen contour (see figure), etc. -R R 0 ε 3. (20 pts) Suppose that f ( z ) = n =0 c n z n is analytic in { z : | z | < R } . Prove that the series φ ( z ) := X n =0 c n z n n ! converges on the whole complex plane. For any fixed r with 0 < r < R , prove that there is a positive constant M (which might depend on
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Unformatted text preview: r ) such that the estimate | φ ( z ) | < Me | z | /r holds for all z ∈ C . 4. (20 pts) Let S denote the strip S = n z ∈ C :-π 2 < Im z < π 2 o . Show that the entire function e e z is bounded on the boundary of S ⊂ C , but it is not bounded on S . 5. (20 pts) Suppose that f ( z ) is continuous on { z : | z | ≤ 1 } - { 1 } and analytic on { z : | z | < 1 } . Prove that if f ( z ) is bounded on { z : | z | < 1 } , then | f ( z ) | ≤ sup | ζ | =1 ,ζ 6 =1 | f ( ζ ) | for each z in { z : | z | < 1 } . Is this inequality valid if f is not assumed to be bounded?...
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