Question

1 10) (20 pts) Evaluate $. 22 +23+3 dz, C is the e ellipse * + y2 = 1

Answer 1

Contour Integral: Given, Pe 224 -dz 2-2 + 72 +3 x? C: + y2 = 1 4 We have to find the value of the integral. First factorize the denominator as follows:

1 22 +7+3 1 22 +6+2 +3 1 2z ( : +3) +1 z +3) 1 ॥ (2z +1) (z+3) - 1 2(z+3) Z- 2 Also C is given as follows: (-D.S, 0)

1 1 Since f(2)= is analytic at 20 - inside or the boundary of the curve, then by the 2(2+3) 2 Cauchy's integral f (2) dz=21if (-o), we get CZ-20 1 Pozz7-3 <dz = $c 2(2+3) dz 1 2+- 2 1 = 2nix 1 2 +3 2πί 5 2πί Hence, the value of the integral is 5