Lobachevsky’s integral formula
Lobachevsky's integral formula states that for an even function f with period π, a certain integral relationship holds. The theorem is useful in Fourier analysis and signal processing, and can be applied even in the special case where f(x) = 1.
Background
- This is a post on *John D. Cook*, a blog by a former math professor and industry researcher who covers applied math, statistics, and programming for practicing engineers and data scientists.
- The "Lobachevsky integral formula" is a theorem from Nikolai Lobachevsky (the Russian mathematician famous for inventing non-Euclidean geometry). It simplifies integrals of even, π-periodic functions — useful in Fourier analysis and signal processing, but not taught in standard calculus.
- The special case f(x) = 1 yields the classic *sinc* and *jinc* integrals, which are foundational in Nyquist sampling theory, antenna design, and image reconstruction.
- For context: readers are expected to already know what an "even function" and "period π" mean; the post is a reference for working mathematicians and engineers.