Consider the function $f:(0,\infty)\to\mathbb{R}$ defined by $$f(k)=\int_{0}^{\frac\pi 2} \sin(\beta) \sin(\pi k\sin(\beta))d\beta$$ $\text{ for all }k>0$. Find $$\min f^{-1}(\{0\})$$ if exists.
I got this problem when I was studying a theoretical physics problem. Existence of a solution can be proved by intermediate value theorem since $f(1)>0>f(2)$. It's better if it's possible to find a general solution to the equation $f(k)=0$ in closed form.
I first wrote $f$ using the Bessel function as $2f(k)=\pi J_1(\pi k)$. But is there a analytical way to calculate the solutions to the equation $J_1(x)=0$ ?
This integral can be expressed after some manipulation can be expressed in terms of a Bessel function:
$$\int_{0}^{\pi/2}\sin x\sin(\pi k \sin x) dx= \frac{\pi}{2}J_1(\pi k).$$
Bessel functions of the first kind are known to have infinite zeros on the real line and the smallest one can be found in various sources to be approximately located at $\pi k =3.8317...$.