Integrating products of BesselJ functions

Question

Supposing $a,b\in\mathbb{R}$, I tried to calculate this kind of integral $$ \int\limits_{-\infty}^\infty \frac{J_1\left(x-a\right)}{x-a}\frac{J_1\left(x-b\right)}{x-b}dx\,, $$ without success. Firstly, I tried to calculate it computing the primitive integral and then determine the limits when $x\to-\infty$ and $x\to\infty$ (because in the case when b=a it was working), but in this case I'm completely stuck.

Answer 1

The Fourier transform of $f(x)=\frac{J_1(x)}{x}$ (which is an even function) is $\widehat{f}(s)=\sqrt{\frac{2}{\pi}}\mathbb{1}_{(-1,1)}(s)\sqrt{1-s^2}$, hence your integral (which is a convolution integral) only depends on $\int_{-1}^{1}(1-x^2)e^{-i(a-b)x}\,dx$ and we have

$$\boxed{ \int_{-\infty}^{+\infty}\frac{J_1(x-a)}{x-a}\cdot\frac{J_1(x-b)}{x-b}\,dx = \color{red}{\frac{8(\sin(a-b)-(a-b)\cos(a-b))}{\pi(a-b)^3}}.}$$ In the limit case $b\to a$ the LHS constantly equals $\color{red}{\frac{8}{3\pi}}$.