A q-Version of the Residue Theorem and Contour Integration: Theory, Computation and Python Applications

Abstract

The classical Residue Theorem is a fundamental result in complex analysis with extensive applications in evaluating contour and improper real integrals. Recent computational developments have demonstrated the effectiveness of combining residue calculus with symbolic computation for numerical verification and visualization. Motivated by these advances, this paper extends the classical residue-theoretic framework to Jackson q-calculus. A Jackson q-Residue-Type Operator and a q-contour-type integral are introduced, leading to a q-Residue Theorem and a corresponding q-Integral Evaluation Formula for the computation of q-integrals. We establish fundamental lemmas to investigate the operator's limiting behaviour and linearity, while the main theorems provide a theoretical foundation for evaluating representative q-integrals. We complement the analytical developments with computational verification in Python, using SymPy, NumPy, mpmath, and Matplotlib for symbolic computation, numerical approximation, and graphical visualisation. Illustrative examples demonstrate the approach's applicability and confirm that the derived q-formulations converge to their classical counterparts as . The results bridge classical residue calculus, quantum calculus, and computational mathematics, providing a foundation for further investigations in q-special functions, q-difference equations, and related areas of mathematical analysis.