Complex Integral Query

Question

Hi so I'm new to Complex integrals and came across this question:

Evaluate $$ \int_C \frac {z^3 + z} {2z + 1} dz $$ where C is the circle $\ \lvert z - 1 \rvert = 1$ in the counterclockwise direction. What's throwing me is the value for C. I know that Cauchy's integral formula is used but then I'm unsure what to do with C. Any help would be greatly appreciated

Answer 1

If $a$ lies inside the contour $C$ you have $$\int_C \frac{f(z)}{z-a} \, dz = 2\pi i f(a).$$ If $a$ lies outside the contour $C$ you have $$\int_C \frac{f(z)}{z-a} \, dz = 0.$$ Write $$\int_C \frac{z^3 + z}{2z + 1} \, dz = \frac 12 \int \frac{z^3 + z}{z + \frac 12} \, dz.$$ Is the point $a = -\frac 12$ inside or outside the contour?

Answer 2

Hint. Recall the Cauchy's integral formula and compare the position of the pole $z=-1/2$ with respect to the curve $C$, that is the circle centred at $1$ and of radius $1$.

Is the pole inside or outside the domain bounded by the circle $C$?