Abstract: This article addresses two examples in quantum physics related to atomic dissipation behavior in structured environments, by utilizing the method of multi-valued function integrals.Both single branch cut and double branch cuts are considered.By appropriately choosing branch cuts and Riemann surfaces, specific procedures for calculating multi-valued function integrals are outlined.This approach can be extended to related areas of research and is expected to contribute significantly to the exploration of quantum phenomena.

Key words: multi-valued functions, Riemann surface, residue theorem