# 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 as $I_2=\dfrac{\pi^2}{16}$ 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, 2015 at 17:16
• @juan My calculations that lead to this integral can be found here wwwsnd.inp.nsk.su/~silagadz/I4.pdf Mar 30, 2015 at 6:26
• But the last integral is not $-\pi^3/12$. I showed that its value is $0$. Look at my answer in mathoverflow.net/questions/200880/interesting-triple-integral
– juan
Aug 11, 2015 at 7:45
• @juan I have accepted you nice answer. Is the integral $$\int\limits_0^\infty \frac{dx}{x} \int\limits_x^\infty \frac{dy}{y}\int\limits_y^\infty \frac{dz}{z} [\sin{x}+\sin{(x-y)}-\sin{(x-z)}-\sin{(x-y+z)}]$$ also zero? Aug 12, 2015 at 6:50