$2\mathrm{.}17$ In section $2\mathrm{.}4\mathrm{.}5$ backward recursion avoided the problems of forward recursion. An$-$

other example is provided by spherical Bessel functions $($of the first kind$)$:

jn$+1($x$)+$ jn$1($x$)=$

$2$n $+1$

x

jn$($x$)(2\mathrm{.}110)$

which is analogous to Bonnet$$s recurrence relation, Eq$.(2\mathrm{.}86).$ Evaluate j$8(0\mathrm{.}5)$ in

the $$naive way$,$ starting from the known functions:

j$0($x$)=$

sin x

x

$,$ j$1($x$)=$

sin x

x

$2$

$$

cos x

x

$(2\mathrm{.}111)$