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Unformatted text preview: r analytic. You may assume that is defined in the normal
. 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
(3 marks) a) Prove that if 2
2sin , then
4 sin (3 marks) 1 b) Show that the real integral given above can be determined from the integral
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.
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
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- Spring '09