I am unable to understand the following statement from my complex analysis textbook:
Let $D$ be an open disc of radius $R$ around $z_0\in \mathbb{C}$, $C$ be any contour lying inside $D$, and $g(z)$ be any function continuous on $C$. If $|g(z)|=1$ for all $z\in D$, then $$\int_C g(z) (z-z_0)^n \ dz = \int_C (z-z_0)^n \ dz$$
I can't see how both of these integrals are equal. It seems like either I am missing something obvious or it is a typo. Please help!
This cannot be true. If the equation holds for every contour $C$ then Morera's Theorem tells us that $(g(z)-1)(z-z_0)^{n}$ is analytic in $D$. But then $g(z)$ itself is analytic (in $D\setminus \{z_0\}$ and hence also in $D$ since it is bounded). But $g(x+iy)=e^{ix}$ is a continuous function which is not analytic and has the property $|g(z)|=1$ for all $z$.