(1) Calculate the residues of the function f(3) =(22+16)(2+ + 25)(22 +4) z 1 at the 3 points z=2i , z=4i and z=5i . (ii)

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1 Calculate The Residues Of The Function F 3 22 16 2 25 22 4 Z 1 At The 3 Points Z 2i Z 4i And Z 5i Ii 1 (80.83 KiB) Viewed 95 times

(1) Calculate the residues of the function f(3) =(22+16)(2+ + 25)(22 +4) z 1 at the 3 points z=2i , z=4i and z=5i . (ii) Use the result of part (i) to calculate the value of the real integral 1-18 dx (x2 +16)(x² +25)(x2 + 4)2 [Hint: First evaluate the integral Sc f(-)dz where C is the simple closed path G+C2 consisting of the straight line segment G from x=-r to x=r on the real axis, and the semi-circle C2 with the parametric equation z = reit for Ostst (where r is any positive real number greater than 5). In the limit as r+00, the integral f(z)dz goes to 21, while 9 the integral along C2 goes to 0, as |S(z)) Sr-8 and |dz| = ire' dt = rdt , and so 15c, ()dz|5ar770 1