Suppose I have a function $G(z)$ with chosen branch cut, say negative real axis, and it's form is known. Now I want to make analytic continuation to $G(z)$ from upper half complex plane to the lower across the branch cut, i.e., I want to get it's formula on the second Riemann sheet. Since the form of $G(z)$ is known, I can calculate the gap between the two sides of the cut$$\operatorname{gap}(\lambda)=\lim_{\eta\to 0^+}{G(\lambda+i\eta)-G(\lambda-i\eta)}$$where $\lambda$ is a real number lying on the branch cut(negative real axis).
So my question is: Is it true that the form of $G(z)$ on the second Riemann sheet can be written as$$G^{\mbox{II}}(z)=G(z)+\operatorname{gap}(z),\ \ \ \ \ \ \operatorname{Im}(z)<0\text{ and }\operatorname{Re}(z)<0$$If not, how can I get the correct formula of $G(z)$ on the second Riemann sheet?