MA530_JAN11 - MATH 530 Qualifying Exam January 2011 G Buzzard S Bell 1(a(10 points Define π f(z:= −π cos2 t dt z − sin t Use the difference

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Unformatted text preview: MATH 530 Qualifying Exam January 2011, G. Buzzard, S. Bell 1. (a) (10 points) Define π f (z ) := −π cos2 t dt. z − sin t Use the difference quotient definition of complex derivative to show that f is holomorphic on C − [−1, 1] and to write f (z ) as an integral. (b) (10 points) Can f (z ) be extended to [−1, 1] to make f (z ) an entire function? If so, describe how to extend f . If not, prove that no such extension exists. 2. (20 points) Let n Pn (z ) = k=0 zk . k! Prove that for a given R > 0, there exists a positive integer N such that if n ≥ N , then Pn (z ) has exactly n zeroes in {z : |z | > R}. 3. (20 points) Suppose that f has a pole at z0 and that g is holomorphic on C − D (so that g has an isolated singularity at ∞). Give necessary and sufficient conditions on the singularity of g at ∞ so that g (f (z )) has a pole at z0 and prove that your conditions are necessary and sufficient. 4. (a) (5 points) Suppose f has a pole of order m at z0 . Use the Laurent series expansion of f around z0 to derive the formula for the residue of f at z0 in terms of derivatives of (z − z0 )m f (z ). (b) (15 points) Compute ∞ 0 ln x dx. + 1)2 (x2 5. (20 points) Prove or disprove that there is a sequence of polynomials {pn (z )}∞ so n=1 that pn (z ) converges uniformly on the unit circle, {z : |z | = 1}, to the function f (z ) = (z )2 . ...
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