On Real Improper Integrals across Isolated Singularities: A Convergence Criterion via Laurent Expansions

This note presents a convergence criterion for the real improper integral ∫f(x)dx when f has isolated singularities x₁ < x₂ < ... < xₙ on the real axis and admits a Laurent expansion near each singularity. The criterion is organized into three cases based on the principal-part coefficients a₋ₙ of the Laurent expansion: (i) If all negative coefficients vanish (a₋ₙ = 0 for all n ≥ 1), every singularity is removable and the integral converges. (ii) If some a₋ₙ ≠ 0 for n ≥ 2 (pole of order ≥ 2), the integral diverges and no finite Cauchy principal value exists. (iii) If only a₋₁ ≠ 0 (simple poles only), the usual improper integral diverges logarithmically, but the Cauchy principal value (PV) exists via symmetric cancellation. The note also presents the author's formulation of the Laurent coefficient formula specialised to the principal part: a₋ₙ = (1/2πi) ∮ f(z) / (z − zⱼ)^(1−n) dz and the vanishing condition a₋ₙ = 0, which directly governs convergence. The original handwritten manuscript is included as an appendix. The underlying mathematical ingredients (Laurent expansion, residue calculus, power test, principal-value theory) are classical. The contribution of this note is the unified three-case formulation, the specialised principal-part notation, and the explicit vanishing condition — formulated and organised by the author. Keywords: improper integral, Laurent expansion, isolated singularity, principal value, residue, convergence criterion, simple pole, removable singularity, independent formulation.