I am going through some of the basic commutation relations in Virasaro Algebra. To obtain the lie algebra for the generators of conformal transformation, I am having some difficulty in complex integration. The statement of my problem is the following: Can we have an identity of the following form analogous to integration by parts in real analysis?
$$\oint_{C} d\omega \,\, \partial_{\omega} \hat{T}(\omega) \,\, \omega^{m+n+2} = -(m+n+2) \oint_{C} d\omega \,\, \hat{T}(\omega) \,\, \omega^{m+n+1}$$
where $m,n \in \mathbb{Z}$