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# Is this function transcendental proof or

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Is this function transcendental? Proof or counterexample. Traditional Problems [10 points each (70 points total)] B1. Assume f ( z ) is meromorphic for all | z | < and satisfies | f ( z ) | ≤ 2 | z | | z - 1 | 3 / 2 . What can you conclude about f ? Justify your assertions. B2. Evaluate A = -∞ cos x x 2 + a 2 dx where a > 0. B3. a) Let f ( z ) be holomorphic in | z | ≤ R with | f ( z ) | ≤ M on | z | = R . Show that | f ( z ) - f (0) | ≤ 2 M | z | R b) Use this to give a proof of Liouville’s theorem.

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2 B4. If f ( t ) is piecewise continuous and uniformly bounded for all t 0, show that for Re { z } > 0 the function (Laplace transform) g ( z ) := 0 f ( t ) e - zt dt is holomorphic for Re { z } > 0. B5. Let f n ( z ) be a sequence of functions holomorphic in the connected open set Ω and assume they converge uniformly on every compact subset of Ω. Show that the sequence of derivatives f n ( z ) also converges uniformly on every compact subset of Ω. B6. Find a conformal map from the unbounded region outside the disks {| z +1 | ≤ 1 }∪{| z - 1 | ≤ 1 } to the upper half plane. B7. Consider the family of polynomials p ( z ; t ) = z n + a n - 1 ( t ) z n - 1 + · · · a 1 ( t ) z + a 0 ( t ) , where the coefficients a j ( t ) depend continuously on the parameter t [0 , 1]. Assume that at t = 0 the polynomial p ( z ; 0) has k zeroes (counted with their multiplicity) in the disk | z - c | < R and has no zeroes on the circle | z - c | = R . Show that for all sufficiently small t the polynomial p ( z ; t ) also has k zeroes in | z - c | < R .

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