Lobachevsky’s integral formula

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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^inftyfrac{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.

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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.

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