Gauss-Legendre principal value integration

As we attempt more sophisticated projects in science and engineering, the mathematical tools we apply to them also become more sophisticated. Because so few problems lend themselves to closed-form solution, we often need to convert formal definitions into practical numerical methods. One such problem deals with the principal value integral, which many students encounter in courses on functions of complex variables. However, the prospect of evaluating such a problem numerically might seem rather daunting; the subject remains outside the treatments of numerical quadrature found in treatises on numerical analysis. This article looks at a simple, efficient procedure for numerically evaluating the Cauchy principal value integral. I_Δ {=P∫_-Δ^Δ[ρ(t+x)/t]dt} is approximated by a Gauss-Legendre quadrature formula.