Applications of Residue Theorem in Modern Mathematical Physics

This paper reviews the residue theorem and demonstrates its efficiency in solving integrals arising in magnetic force, topological phase transitions, and fidelity susceptibility. It highlights the theorem as a bridge between mathematical theory and physical applications.

X. Ren· Theoretical and Natural Science· 2026· DOI 10.54254/2753-8818/2026.dl33834

The core problem

The residue theorem is a cornerstone of complex analysis with wide-ranging applications in mathematical physics. Despite its established theoretical foundation, it continues to offer significant convenience and efficiency in modern physics. This paper aims to showcase its practical utility by applying it to various physical problems, including magnetic force calculations, topological phase transitions, and fidelity susceptibility. The focus is on one-dimensional models to provide clear derivations, transforming complex real integrals into contour integrals that can be solved analytically via residue calculus. The key step involves identifying all singularities, which enables straightforward evaluation. The results underscore the residue theorem as a powerful and rigorous method for tackling challenging physical integrals, particularly when compared to traditional definite integral techniques. By bridging mathematical theory and physical applications, this unified framework facilitates conceptual understanding and promotes broader adoption in related research.

The methodology centers on the residue theorem from complex analysis. For a function f (z) analytic inside and on a simple closed contour C, except for isolated singularities z_{k} inside C, the theorem states: \oint_{C} f (z) d z = 2 π i k \sum Res (f, z_{k}) where Res (f, z_{k}) is the residue of f at z_{k} . To apply this to physical integrals, we extend the integrand to the complex plane, choose an appropriate contour (often a semicircle in the upper half-plane for integrals over the real line), and evaluate the residues at the enclosed singularities. The contour is then deformed or closed such that the integral over the added path vanishes or is related to the original integral. This approach converts difficult real integrals into sums of residues, yielding analytical results. The paper applies this method to three specific physical contexts: (1) magnetic force calculations, (2) topological phase transitions, and (3) fidelity susceptibility. In each case, one-dimensional models are used to keep derivations transparent and to highlight the general applicability of the technique.

Introduction

The residue theorem is a cornerstone of complex analysis with wide-ranging applications in mathematical physics. Despite its established theoretical foundation, it continues to offer significant convenience and efficiency in modern physics. This paper aims to showcase its practical utility by applying it to various physical problems, including magnetic force calculations, topological phase transitions, and fidelity susceptibility. The focus is on one-dimensional models to provide clear derivations, transforming complex real integrals into contour integrals that can be solved analytically via residue calculus. The key step involves identifying all singularities, which enables straightforward evaluation. The results underscore the residue theorem as a powerful and rigorous method for tackling challenging physical integrals, particularly when compared to traditional definite integral techniques. By bridging mathematical theory and physical applications, this unified framework facilitates conceptual understanding and promotes broader adoption in related research.

The residue theorem emerges as a highly efficient and rigorous tool for solving intricate physical integrals in both classical electromagnetism and quantum condensed matter physics. Its advantage over traditional definite integral calculation methods lies in its ability to convert real integrals into contour integrals, which can often be evaluated exactly by summing residues. This not only saves computational effort but also provides deeper analytical insight. The unified framework offered by the residue theorem bridges mathematical theory and physical applications, making relevant concepts more accessible and easier to grasp. For instance, in topological phase transitions, the theorem connects the mathematical notion of residues to physical observables like topological invariants, enhancing understanding. Similarly, in fidelity susceptibility, it links complex analysis to quantum critical phenomena. The paper's use of one-dimensional models ensures clarity and straightforwardness, making the method approachable for a broad audience. The findings suggest that the residue theorem will continue to be a valuable asset in theoretical physics, promoting more extensive use in related research. Future work could extend this approach to higher-dimensional systems and more complex singularities.

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