Riemann-Roch in the version I know it:

Let $(E,\bar{\partial})$ be a holomorphic bundle over a compact Riemann surface $M$. Then $$index(\bar{\partial}) = deg E -(g-1)rank E$$

Here $index(\bar{\partial}) = dim H^0 (E) - dim H^0(KE^*)$.
This version arises, if you proof the fundamental theorem for elliptic operators ($\bar{\partial}$ is an elliptic operator, and Serre duality states, that two $\bar{\partial}$ operators on a complex vector bundle have the same index). (By the way: This is just a reformulation of RR, stated with divisiors (use Kodaira and Chow (?) )

Riemann-Roch in the version I know it:

Let $(E,\bar{\partial})$ be a holomorphic bundle over a compact Riemann surface $M$. Then

$$\operatorname{index}(\bar{\partial}) = \deg E -(g-1)\operatorname{rank} E.$$

Here $\operatorname{index}(\bar{\partial}) = \dim H^0(E) - \dim H^0(K\otimes E^*)$.

This version arises, if you proof the fundamental theorem for elliptic operators ($\bar{\partial}$ is an elliptic operator, and Serre duality states, that two $\bar{\partial}$ operators on a complex vector bundle have the same index). (By the way: This is just a reformulation of RR, stated with divisiors (use Kodaira and Chow (?) )