find the integral?

Question

∫C(0,1) of ((z^2-3z+1)/(z(z^2-3)))dz
calculate the integral and find all the redisudes.

*C(0,1) is the circle center at 0 with radius 1.

Answer 1

Your function is analytic function with roots at origin and other roots outside of C(0,1) circle. Therefore according to residual theorem your integral = 2*pi*i* Res(0,0). Residuum at origin = Res(0,0) = lim|z ->0|(z*f(z)) where f(z) = (z^2-3z+1)/(z(z^2-3)). Therefore Res(0,0) = -1/3, and the value of your integral = - 2*pi*i/3, where pi = 3.14... and i = sqrt(-1).