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Math 617 Theory of Functions of a Complex Variable Fall 2015 Final Examination Instructions Solve six of the following eight problems (most of which are taken from the textbook). 1. State the following theorems (with precise hypotheses and conclusions): Riemann’s theorem on removable singularities, the homology version of Cauchy’s theorem, and one of Picard’s theorems. ( √ )6 −1 − ๐ 3 2. Find both the real part and the imaginary part of . 2 3. Let ๐บ be the complex plane slit along the negative part of the real axis: namely, ๐บ = โ โงต { ๐ง ∈ โ โถ ๐ง ≤ 0 }. Let ๐ be a positive integer. There exists an analytic function ๐ โถ ๐บ → โ such that ๐ง = (๐ (๐ง))๐ for every ๐ง in ๐บ. Find every possible such function ๐ . ๐๐๐ง ๐๐ง when ๐พ is the unit circle traversed once counterclockwise ∫๐พ ๐ง2 [that is, ๐พ(๐ก) = ๐๐๐ก and 0 ≤ ๐ก ≤ 2๐]. 4. Evaluate 5. Let ๐ be a positive real number, let ๐ be a real number, and let ๐พ be a continuously differentiable curve in โ โงต {0} that begins at the point 1 and ends at the point ๐๐๐๐ . Prove the existence of an integer ๐ such that 1 ๐๐ง = ln(๐) + ๐๐ + 2๐๐๐. ∫๐พ ๐ง 6. Suppose ๐ (๐ง) = 1 . Find the singular part of ๐ at each singular point. 1 − ๐๐ง 7. Let ๐บ be a bounded open set. Suppose the function ๐ is continuous on the closure of ๐บ and analytic on ๐บ. Additionally, suppose there is a positive constant ๐ such that |๐ (๐ง)| = ๐ for every ๐ง on the boundary of ๐บ. Prove that either ๐ is a constant function, or ๐ has at least one zero in ๐บ. 8. Apply Rouché’s theorem to prove the fundamental theorem of algebra (every polynomial of degree ๐ has ๐ zeroes, counting multiplicity). December 16, 2015 Page 1 of 1 Dr. Boas