How to integrate double integral
$$\int_{0}^{\infty}\!\int_{0}^{2\pi}\ \frac{1}{2}\left(\frac{\partial}{\partial x}-\frac{\partial}{\partial y}\right)g_m \bar{g_n} , d\theta dr$$ where $$g_a=(x+iy)^a$$ . I do not know how to differentiate first part of integral.