Question

2x 1 Use a contour to evaluate dᎾ (2+0 -cos)

Answer 1

Salection! 25 do (2+ coso) 2 but 2 - eio so that dz iz = do I: dz - 22 do Jo (2+coso)? I to z{2+{}}{2+3] .-41 z dz Ic (22+42 +1)2 - 41 z da Jc (z? +42+1) 2 -41 f f(2) dz where C is the unit circle 121=1 Poles of order two of fcz) are given By the Roots of z t 42 t 120 z = - 3 + 13 d. z = -2713 Z=-2 +53 = - B

Sine it follows IBILI and L. B = TRI< of order too Hold pale Hence & is only incide c. z How 4i f(2)= 4i2 (2-0) 2 (2-3)2 cz) .(2-)? where 4 iz $CZ) = (2-32 Residue at the double pole z=a is $(0) - -41 (z-ß) 22.02-B) (2-3)4 Z=2 -41 -$-22 -33 +41 (O+B) (4-B)3 X 62 - -41 (-4) 0 (4-193/2 gi 4333/2

Residene theorem Hene By Cauchy Residene do ati Levy 2i (3) 3/2 (2+coso? 47 (3) 3/2 Answer 27 do 4 T (2+ cos o² (3) 3/2 Answer

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