I have an integral $\int_0 ^a\int_0 ^b\int_0 ^a\int_0 ^b \sin(x)\sin(\bar{x})\sin(y)\sin(\bar{y})f(x,\bar{x},y,\bar{y})dxdyd\bar{x}d\bar{y}$ where $f= \dfrac{\sin(\sqrt{(x-\bar{x})^2+(y-\bar{y})^2})}{(x-\bar{x})^2+(y-\bar{y})^2}$
under the transformation $\bar{x}=x+u$ and $\bar{y}=y+v$ $\Rightarrow$ $f$ becomes $f(u,v)$ but the limits of integration with respect to $u$ and $v$ change.
Considering only the x part, $\int_{-x} ^{a-x}\int_0 ^a \sin(x)\sin(x+u)f(u,v)dxdu$ integrating this with respect to u analytically is difficult while with respect to x is easier. But the $u$ limit have $x$ terms can I by anyway, make the limits of $u$ as $0$ to $a$ but yet carry a simpler integration with respect to $x$, changing the limits of $x$ would cause no harm.
