# Multiple Integral (American Mathematical Monthly problem 11621 and its generalization)

AMM problem 11621 asks to calculate the integral $$I_2=\int\limits_{-\infty}^{\infty}ds_1\int\limits_{-\infty}^{s_1}ds_2 \int\limits_{-\infty}^{s_2}ds_3\int\limits_{-\infty}^{s_3}ds_4 \;\cos{(s_1^2-s_2^2)}\;\cos{(s_3^2-s_4^2)}.$$ Its multi-dimensional generalization was considered in http://arxiv.org/abs/1201.1975 where the integral was calculated thanks to some physics input. However the simplicity of the result suggests that there should exist some direct method of calculation without any reference to the Landau-Zener connection. Up to now, I was unable to tackle the general case. Maybe someone could indicate any relevant reference or idea? I found some references indicated in https://wwwsnd.inp.nsk.su/~silagadz/I2.pdf which facilitates the calculation of $I_2$, however, these tricks do not work in general case unless we can prove the equality (which should be true) $$\int\limits_{-\infty }^{\infty }ds_{1}\int\limits_{-\infty}^{s_{1}}ds_{2}\cdots \int\limits_{-\infty }^{s_{2n-1}}ds_{2n}\;\cos { (s_{1}^{2}-s_{2}^{2})}\;\cdots \cos {(s_{2n-1}^{2}-s_{2n}^{2})}=$$ $$\int\limits_{-\infty }^{\infty }ds_{1}\int\limits_{-\infty}^{s_{1}}ds_{2}\cdots \int\limits_{-\infty }^{s_{2n-1}}ds_{2n}\;\cos { (s_{1}^{2}-s_{2}^{2}+s_{3}^{2}-s_{4}^{2}+\cdots +s_{2n-1}^{2}-s_{2n}^{2})}.$$

And one more question. In one of my attempts to directly calculate $I_4$ I stumbled on an alleged identity $$\int\limits_0^\infty \frac{dx}{x} \int\limits_0^x \frac{dy}{y} \int\limits_0^y \frac{dz}{z} [\sin{x}+\sin{(x-y)}-\sin{(x-z)}-\sin{(x-y+z)}]= -\frac{\pi^3}{12}.$$ How this identity can be proved?

-
The integral is not absolutely convergent. I understand its value as the limit for $t\to\infty$ of the integral you get substituting $\infty$ by t in the integral in $x$. For this case I have some computations that pointed to a value $0$. It is possible that some sign is wrong in the integrand? –  juan Mar 29 at 17:16
@juan My calculations that lead to this integral can be found here wwwsnd.inp.nsk.su/~silagadz/I4.pdf –  Zurab Silagadze Mar 30 at 6:26