Question Five (1) Calculate the residues of the function f(2)= + 4)(22+16)(2+ +932 1 at the 3 points z=2i, z=3i and z=4i

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Question Five 1 Calculate The Residues Of The Function F 2 4 22 16 2 932 1 At The 3 Points Z 2i Z 3i And Z 4i 1 (117.73 KiB) Viewed 74 times

Question Five (1) Calculate the residues of the function f(2)= + 4)(22+16)(2+ +932 1 at the 3 points z=2i, z=3i and z=4i. (ii) Use the result of part (i) to calculate the value of the real integral dx 1=50 (+ 4)(x2 +16)(x2 +932 = + [Hint: First evaluate the integral [cS(z)dz where C is the simple closed path G+C2 consisting straight line segment (from x=-r to x=r on the real axis, semi-circle C2 with the parametric equation z=relt for Ostst (where r is any positive real number greater than 4). In the limit as r+00, the integral So, S()dz goes to 21, while the integral along C2 goes to 0, as |f(z)| Sr-8 and |dz| = ire" dt |= rdt , and so 15c, 5(=)d=15ar7 ] →0