I have a question about Bessel functions, I know that the integral representation of the Bessel function of the first kind is:
$J_n(x) = \frac{1}{\pi}\int_{0}^{\pi} \cos(x\sin(\theta)-n\theta) \,d\theta$
Now, what is this integral?
$\frac{1}{\pi}\int_{0}^{\pi} \sin(x\sin(\theta)-n\theta) \,d\theta$
Is it related to Bessel functions? Or is it something else entirely?
Sidenote: I’ve also seen this:
$J_n(x) = \frac{2}{\pi}\int_{0}^{\frac{\pi}{2}} \cos(x\sin(\theta)-n\theta) \,d\theta$
What is the difference between this integral and the one I wrote above?
Thank you.