Abstract: The fidelity and its derivative, the fidelity susceptibility (FS), have emerged as pivotal concepts in quantum information theory and have garnered significant attention as robust probes for detecting quantum phase transitions. However, the derivation of analytical solutions for the FS remains a challenge, primarily due to the paucity of quantum integrable models. To address this issue, this paper employs the one-dimensional XY model as a paradigmatic example. Initially, a series of parameterized integrals are meticulously constructed, and their analytical expressions are adeptly derived through the application of the residue theorem. Subsequently, the intricate relationship between the FS and these parameterized integrals is elucidated across different phase transition processes, culminating in the derivation of a closed-form expression for the FS. This study underscores the crucial role of the FS in identifying quantum phase transitions and accentuates the remarkable efficacy and superiority of the residue theorem in tackling complex physical problems.

Key words: residue theorem, XY model, fidelity susceptibility, quantum phase transition