MA530_JAN05

MA530_JAN05 - D = z | z |< 1 Im z> onto the unit disc...

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QUALIFYING EXAMINATION JANUARY 2005 MATH 530 - Prof. Lempert Each problem is worth 5 points. 1. Suppose that f is holomorphic and nonconstant in a disc | z - a | <r . Show that lim z a log | f ( z ) - f ( a ) | log | z - a | exists, and is a nonnegative integer. 2. Prove that if g has a pole and h has an essential singularity at c then gh has an essential singularity at c . 3. Evaluate Z -∞ e ix ( x 2 +1)( x 2 +4) dx (and show your work). 4. Show that if a holomorphic function φ :C C satisfies | φ ( z ) |≤ e | z | for all z ,then there is a c> 0 such that | φ 0 ( z ) |≤ ce | z | for all z . 5. Construct a one–to–one holomorphic map from the half–disc
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Unformatted text preview: D = { z : | z | < 1 , Im z > } onto the unit disc U (i.e., a biholomorphic map D → U ). 6. Let P ( z ) = z n + a 1 z n-1 + ··· + a n be a polynomial and 0 < θ < π/ (2 n ). Show that e P ( z ) → ∞ , as z → ∞ , | arg z | < θ, and e P ( z ) → , as z → ∞ , | arg z-π/n | < θ. 7. Prove that if Q is any polynomial, then | Q ( z )-1 z | ≥ 1 for some z with | z | = 1....
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