MAT290H1F_FinalExam_2009

Clearly state your reasons why the function is or is

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Unformatted text preview: r analytic. You may assume that is defined in the normal way, i.e., . Page 13 of 17 (2 marks) 5. d) (i) Calculate the result of 1 √3 . You must show all your complex arithmetic to receive credit for this question. However, you should verify your answer with your calculator. (3 marks) (ii) Determine the result of 1 (2 marks) (iv) Give a brief explanation of why your result from part (ii) is not equal to 1 √3 √3 . ? Page 14 of 17 6. All parts of this question deal with the evaluation of the integral 1 (3 marks) a) Prove that if 2 2sin , then 2 4 sin (3 marks) 1 b) Show that the real integral given above can be determined from the integral 2 where 4 is the closed contour defined by | | 1 1 travelled in the counterclockwise sense. Page 15 of 17 (6 marks) 6. c) Identify all the pole(s) and zero(s) of the integrand given in part (b), i.e., 4 1 Indicate the type of each pole and zero, and draw the placement of these pole(s) and zero(s) in the complex plane and include a sketch of the contour, , described in part (b). You must show your work in determining the roots of the denominator. (8 marks) d) Use Cauchy’s residue theorem to evaluate the integral. 1 2 2sin Page 16 of 17 f(t) f (t ) F(s) 1 f (t ) F(s) 0 cos 1 sin 2! cosh ! sinh 1 1 cos sin Page 17 of 17 F (s ) 1 1 0...
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