<s>[INST] 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?
 [/INST] 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
$$
J_{1/2}(x) = \sqrt{\frac{2}{\pi x}} \sum_{n=0}^{\infty} \frac{(-1)^n}{\Gamma(2n+2)} x^{2n+1} = \sqrt{\frac{2}{\pi x}} \sin x.
$$
</s>