$\Gamma$ refers to a once continuous counterclockwise closed circle with center $\xi =2$ and with a positive radius $r=3$. I need to determine the following integral $\frac{1}{z+2}$.
Why is $\int_{\Gamma } f \left ( z \right )dz=0?$
I know that is something connected with holomorphic function and line integral but still can't find the rules how to calculate integral of function when I have specific value of radius and center?
If someone could help me, I would be really thankful.
On
$\log(z+2)$ is holomorphic in $\mathbb C \setminus (-\infty,-2]$ (which contains the circle in question) and is an antiderivative of $1/(z+2)$ in this region. To evaluate the integral, you simply subtract the values of the antiderivative at the endpoints. Since this is a closed curve, the endpoints are the same, and we get $0$ for the integral
The only singularity of $f$ is at $z = -2$, which is not in $\Gamma$ and neither in its interior. So $f$ is holomorphic there. Since $\Gamma$ is closed, by Cauchy-Goursat follows that $\int_\Gamma f(z)\,{\rm d}z = 0$.