Integral resulting in Bessel function

[wnvl]

Homework Statement

$$\int_{0}^{\infty} \sin \left(x\right) \sin \left(\frac{a}{x}\right) \ dx = \frac{\pi \sqrt{a}}{2} J_{1} \left( 2 \sqrt{a} \right)$$ where $$J_{1}$$ is the Bessel function of the first kind of order 1.

Homework Equations

The Attempt at a Solution

Some calculations I did already

$$\int_{0}^{\infty} \sin \left(x\right) \sin \left(\frac{a}{x}\right) \ dx= \int_{0}^{\infty} \sum_{k=0}^{\infty }(-1)^{k}\frac{x^{2k+1}}{2k+1!} \cdot \sum_{l=0}^{\infty }(-1)^{l}\frac{a^{2l+1}x^{-2l-1}}{2l+1!} \ dx$$

$$=? \int_{0}^{\infty} \sum_{l=0}^{\infty } \sum_{k=0}^{\infty }(-1)^{k+l}\frac{x^{2(k-l)}}{(2k+1)!(2l+1)!} a^{2l+1} \ dx$$


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$$\frac{\pi \sqrt{a}}{2}J_{1}(2\sqrt{a})=\frac{\pi \sqrt{a}}{2} \sum_{l=0}^{\infty}\frac{(-1)^l}{2^{2l+1}l!(1+l)!} 2^{l+\frac{1}{2}}a^{l+\frac{1}{2}}$$
$$=\pi \sum_{l=0}^{\infty}\frac{(-1)^l}{2^{l+\frac{3}{2}}l!(1+l)!} a^{l+1}$$

I put ? because I think this step is not allowed because of the singularity of $$\sin \left(\frac{a}{x}\right)$$ at x=0. Can someone confirm if this equality is true?

 

[kurt.physics]

I am stuck at this point. I also would appreciate if someone could give a hint about how to get the solution from here.