confusion about stokes theorem

Question

Consider the open unit disk $\mathbb{D}$ in complex plane $\mathbb{C}$, let $\mathbb{U}$ be the open disk of the origin with radius $\frac{1}{2}$, consider the manifold with boundary $\mathbb{D}-\mathbb{U}$ and a holomorphic form $\omega=\frac{1}{z}dz$ on it where $dz=dx+idy$. Then $d\omega$ is zero on $\mathbb{D}-\mathbb{U}-\partial\mathbb{U}$ since $\frac{1}{z}$ is holomorphic on it hence by stoke's theorem $$ 0=\int\int_{\mathbb{D}-\mathbb{U}-\partial\mathbb{U}} d\omega=-\int_{\partial\mathbb{U}}\omega=-\int_{\partial\mathbb{U}}\frac{1}{z}dz $$ But we know the right side is $-2\pi i$, what is wrong in this computation?

Answer 1

$$\int_{1/2 < \lvert z\rvert < 1} d\omega= \int_{\lvert z\rvert = 1} \omega - \int_{\lvert z\rvert = 1/2} \omega=-2\pi i+2\pi i=0.$$