Bessel function to $\sin(kr)$ $J_{\frac{1}{2}}(kr)=\frac{\sqrt{\frac{2}{\pi }} \text{Sin}[\text{kr}]}{\sqrt{\text{kr}}})$ This can be easily obtained by Mathematica,
How to do the details?

One way is to use the series definition of the Bessel function together with the duplication formula for the Gamma function.
For $x \geq 0$ we have
$$
\begin{align}
J_{1/2}(x) &= \sqrt{\frac{x}{2}} \sum_{n=0}^{\infty} \frac{(-1)^n}{\Gamma(n+1)\Gamma(n+1+1/2)} \left(\frac{x}{2}\right)^{2n} \\
&= \sqrt{\frac{2}{x}} \sum_{n=0}^{\infty} \frac{(-1)^n}{\Gamma(n+1)\Gamma(n+1+1/2)} \left(\frac{x}{2}\right)^{2n+1}.
\end{align}
$$
Now
$$
\Gamma(n+1)\Gamma(n+1+1/2) = 2^{-1-2n} \sqrt{\pi} \,\Gamma(2n+2)
$$
by the duplication formula, so
$$
J_{1/2}(x) = \sqrt{\frac{2}{\pi x}} \sum_{n=0}^{\infty} \frac{(-1)^n}{\Gamma(2n+2)} x^{2n+1} = \sqrt{\frac{2}{\pi x}} \sin x.
$$
