The evaluation of a definite integral by the method of brackets illustrating its flexibility
Ivan Gonzalez, John Lopez Santander, Victor H. Moll
TL;DR
This paper investigates the method of brackets as a flexible, rule-based approach to evaluating definite integrals on the half-line, using the integral $I(\alpha,\beta)=\int_0^{\infty}\int_0^{\infty} x^{\alpha-1} y^{\beta-1} \mathrm{Ei}(-x^{2}y) K_{0}\left(\frac{x}{y}\right) dx dy$. It showcases a spectrum of evaluation strategies: direct bracket-based proofs, non-classical divergent/null series, Mellin transforms, contour/Mellin-Barnes representations, and mixed methods, all converging to the same closed form $I(\alpha,\beta) = -\frac{1}{12} \frac{ \Gamma^{2}\left( \frac{\alpha+\beta}{3} \right) \Gamma^{2}\left( \frac{\alpha-2\beta}{6} \right) }{4^{(2\beta-\alpha)/6} \Gamma\left( \frac{\alpha+\beta}{3} + 1 \right)}$. The work highlights the method's ability to handle integrands with logarithmic singularities (from $K_{0}$ and $\mathrm{Ei}$) and its versatility in combining classical, non-classical, and complex-analytic techniques to produce consistent results with diverse representations.
Abstract
The method of brackets is an procedure to evaluate definite integrals. It is based on a small number of operational rules. The flexibility of this method is illustrated with the evaluation of an integral involving the Bessel K0 function and the exponential integral. Several proofs are presented.