From the Dirichlet Integral to Lobachevsky's Formula: A Formalization in Lean 4

Abstract

We formalize the Dirichlet integral and several of its classical applications in the Lean~4 proof assistant. Since the sinc function is not Lebesgue integrable on the positive half-line, the Dirichlet integral must be represented as the limit of integrals over bounded intervals. To avoid the difficulty of removing an exponential factor from a conditionally convergent integral, we instead pass through the absolutely integrable function \(\operatorname{sinc}^2\). We evaluate its integral by differentiation under the integral sign and dominated convergence, and then recover the Dirichlet integral from an identity between truncated integrals. Using these results, we formalize the convergence of the Dirichlet cutoff to the Heaviside function and derive several quadratic and bilinear trigonometric integral identities. Finally, we formalize Lobachevsky's integral formula for continuous periodic functions satisfying a reflection symmetry, using the density of cosine polynomials obtained from Mathlib's Fourier analysis on the additive circle.