Do Bessel functions have any "nice" input/output pairs?

Question

By analogy with the sine function, we have the following:

$$\begin{align}\sin(0) &= 0, \\ \sin\left(\frac{\pi}{2}\right) &= 1, \\ \sin(\pi) &= 0,\end{align}$$

etc.

I'm wondering if Bessel functions have anything similar.

Answer 1

For example, consider that

$$J_{1/2}(x) = \sqrt{\frac{2}{\pi}} \frac{\sin{x}}{\sqrt{x}}$$

so that

$$J_{1/2} \left ( \frac{\pi}{2}\right ) = \frac{2}{\pi}$$