Lobachevsky’s Formula for Dirichlet Integrals: Beauty in Improper Integration

Happy Mathematician’s Day

For the second year now, December 1st is celebrated as Mathematician’s Day in honor of the birthday of Nikolai Ivanovich Lobachevsky.

While Lobachevsky is best known as the creator of non-Euclidean geometry, his contributions to science, education, and academic life extend far beyond that revolutionary discovery.

The Lobachevsky Dirichlet Integral Formula

One important but little-known achievement is the Lobachevsky formula for Dirichlet integrals.

This formula allows us to evaluate certain classes of improper integrals and holds fundamental significance in distribution theory.

For a continuous function f(x) satisfying the conditions of π-periodicity f(x+π) = f(x) and symmetry f(π−x) = f(x), the Lobachevsky formula states:

∫₀^∞ (sin x / x) f(x) dx = ∫₀^(π/2) f(x) dx

Why the Lobachevsky Integral Formula Matters

This elegant formula connects an improper integral over an infinite interval with an ordinary integral over a finite segment. Modern mathematicians continue to generalize and extend this formula to more complex cases — a testament to its foundational importance in mathematical analysis.

A Beautiful Special Case

To obtain a striking result, it is enough to take f(x) = 1. Try it yourself — or take a look at the illustration at the top of the post.