Evaluate∮_C (f(z))dz where f (z) is given by the following: f(z)=1/(2z+1)

Let C be the unit square with diagonal corners at −1 − i and 1 + i .
Evaluate∮_C (f(z))dz where f (z) is given by the following: f(z)=1/(2z+1)

This time, the function is analytic for all z except at z = -1/2, which is inside C.

By Cauchy's Integral Theorem,
∮dz/(2z + 1)
= (1/2) ∮1 dz/(z + 1/2)
= (1/2) * 2πi * 1 {at x = -1/2}
= πi.

I hope this helps!