Question

where the $\gamma$ is the circle at the origin travelles counter clockwise find $\int_{\gamma }$dz

Answer 1

Let f (z) = 1/(z - a) (z -b) Case I If a and b lies outside the circle Squareroot. Then f is analytic in squareroot Therefore integral_r 1/(z - a) (z - b) dz = 0 Case II If a lies inside squareroot and b does not. then f(z) has simple pole at z = a Res f(z)_z = a = lim_z rightarrow a (z - a) f(z) = lim_z rightarrow a (z - a) 1/z - b = 1/a - b (a notequalto b) integral_r 1/(z - a) (z - b) dz = 2 pi 2 1/a - b (by Cauchy residue theorem) Case III If b lies inside squareroot and a does not. Then f(z) has simple pole at z = b. Res f(z) = lim_z rightarrow b = lim_z rightarrow b f(z) lim_z rightarrow b 1/z - a = 1/b - a (b notequalto a) By Cauchy residue theorem integral_r 1/(z - a)