MAT290H1F_FinalExam_2009

This path starts at 4 then travels through the origin

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Unformatted text preview: passes back through the origin to , then goes through 4 the origin before returning to C4 −2 2 −4 Page 11 of 17 x (4 marks) 5. a) State whether or not the following statements about the Taylor series expansion of cos are true or false. Briefly justify your answer with sound mathematical reasoning, or calculation. True False (i) The radius of convergence is R = π/2, for the expansion around z = π. True (ii) The series cannot be expanded about z0 = π/2. False True False (iii) For the same point of expansion, z0, we can find other power series to represent the same function in |z − z0| < R. True False (iv) The Maclaurin series expansion does not contain any odd powers of z. Page 12 of 17 (2 marks) 5. b) The Laurent series of a function valid for 0 | 1| is given by 1 1 1 3! 5! 7! From this series what can you conclude about the type of pole of what is Res( ,1)? 1 (7 marks) at 1 and 5. c) For the function 3 3 3 3 , determine where is differentiable and where it is analytic. Clearly state your reasons why the function is or is not differentiable and/o...
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