Lobachevsky’s integral formula

Let f be an even function with period π. Then the following remarkable theorem by Lobachevsky holds.

$\int_0^\infty \frac{\sin^2 x}{x^2} f(x) \, dx = \int_0^\infty\frac{\sin x} x f(x) \, dx = \int_0^{\pi/2} f(x) \, dx$

This theorem is useful in Fourier analysis and signal processing. It’s useful to know even in the special case f(x) = 1.

For a “jinc” analog, see this paper.

***

Every time I see the name Lobachevsky I think of Tom Lehrer’s song about him. You can find the words here and the audio here.

Related posts