For (10.59.1) suppose first $b\ne 0$. The left-hand side is
$2\mathrm{i}{\int}_{0}^{\mathrm{\infty}}\mathrm{sin}\left(bt\right){\U0001d5c3}_{n}\left(t\right)dt$ or
$2{\int}_{0}^{\mathrm{\infty}}\mathrm{cos}\left(bt\right){\U0001d5c3}_{n}\left(t\right)dt$ according as $n$ is odd
or even, see (10.47.14). Next, apply (10.22.64)
with $a=1$, $\mu =\frac{1}{2}$ or $-\frac{1}{2}$, and subsequently replace
$2n+1$ or $2n$ by $n$. For ${J}_{\pm (1/2)}\left(bt\right)$ and
${J}_{n+(1/2)}\left(t\right)$ we have (10.16.1) and
(10.47.3); also the function ${}_{2}{}^{}F_{1}^{}$ is interpreted as a
Legendre polynomial for both odd and even $n$ via (14.3.11),
(14.3.13), and (14.3.14).
When $b=0$, use (10.22.43), (10.47.3), and also
${P}_{n}\left(0\right)={(-1)}^{\frac{1}{2}n}{\left(\frac{1}{2}\right)}_{\frac{1}{2}n}/(\frac{1}{2}n)!$ or $0$, according
as the nonnegative integer $n$ is even or odd; see (14.5.1) and
§5.5.

where ${P}_{n}$ is the Legendre polynomial (§18.3).

For an integral representation of the Dirac delta in terms of a product of
spherical Bessel functions of the first kind
see §1.17(ii), and for a generalization
see Maximon (1991).

Additional integrals can be obtained by combining the definitions
(10.47.3)–(10.47.9) with the results given in
§10.22 and §10.43. For integrals of products see also
Mehrem et al. (1991).