List of practice Questions

Let $f(z)$ be continuous in a simply connected region $G$ and suppose $\oint_c f(z)dz = 0$ around every simple closed curve $c$. Then

Consider a function \(f(z)=z^2+z+1\), where \(z\in\mathbb{C}\) is a complex variable. A simple closed contour \(\gamma\) in the \(z\)-plane encloses the point \(z=1+0j\).
The value of the integral
\[\oint_{\gamma}\frac{f(z)}{z-1}\,dz\]
is ______.