is n!*n! somehow related to (2n)!?
I'm trying to prove that (sorry for the bad formatting...)
integral(cos(x*sin(theta)), theta, 0, pi/2) = c*J0(x)
J0(x) is a bessel function of order zero.
What I did:
I turned the integral, or more specifically, the cos expression into the taylor series expression and also turned J0(x) into a series expression and am trying to cancel both sides. If n!*n! is equal to (2n)! or is somehow related by a constant or something it might be helpful.
Otherwise, if you all can think of a more elegant way to prove it, I'm open to suggestions!
No, n!*n! is not equal to nor simply related to (2n)!. Just look at some values: if n= 2, 2!*2!= 4 while (2(2))!= 4!= 24. If n= 3, 3!*3!= 36 while (2(3))!= 6!= 720. You should be able to show that n!n! always divides (2n)! but there is no closer relationship than that.