Chapter 8.8, Problem 26E

The Bessel functions of order $v=n+1/2$, $n$ any integer, are related to the spherical Bessel functions. Use relation (33) and the results of Problem 25 to show that such Bessel functions can be represented in terms of $\sin \text{x}$, $\cos \text{x}$, and powers of $\text{x}$. Demonstrate this by determining a closed form for $J_{-3/2}\left(x\right)$ and $J_{5/2}\left(x\right)$.

$J_{v+1}\left(x\right)=\frac{2v}{x}{J}_v\left(x\right)-J_{v-1}\left(x\right)$, $\left(33\right)$

25. Show that

$J_{1/2}\left(x\right)={\left(2/\pi x\right)}^{1/2}\sin x$ and

$J_{-1/2}\left(x\right)={\left(2/\pi x\right)}^{1/2}\cos x$.