I would like to show that for the Bessel functions, we have:
$J_{-m}(x) = (-1)^m J_m(x)$.
This can be obtained by using the definition of the Bessel functions as
$J_m(x) = \big(\frac{x}{2}\big)^m \sum_{k = 0}^{\infty} \frac{(-1)^k}{k! (k + m)!} \big(\frac{x}{2}\big)^{2k}$.
Now, I have come across the integral representation of the Bessel functions which reads
$J_m(x) = \frac{1}{\pi} \int_0^\pi d\theta \cos(x \sin\theta - m \theta)$.
Is it possible to obtain the negative integer order feature of the Bessel functions from this integral?
Edit
This problem can be solved by a change of variable $\theta = \pi - t$ in the expression for $J_{-m}(x)$.