# multivariable calculus – Interchange of order of integration for \$int_0^infty y^{s-1} e^{-ay} int_0^infty frac{sin(2yx)cos(pi x^2)}{sinh(pi x)}dx dy \$

I have been trying to interchange the order of integration for the integral: $$int_0^infty y^{s-1} e^{-ay} int_0^infty frac{sin(2yx)cos(pi x^2)}{sinh(pi x)}dx dy$$

I am unable to find a proof for the same. Using Mathematica I was able to see that the values are the same even when we interchange the order of integration upto some $$s,a in mathbb{R}$$ with $$a>0$$.

But for larger values of $$s$$, mathematica says it is divergent, so I am interested in the condition on $$s,a$$ such that we can interchange the order of integration.

Any help is highly appreciated.