# §10.59 Integrals

 10.59.1 $\int_{-\infty}^{\infty}e^{ibt}\mathop{\mathsf{j}_{n}\/}\nolimits\!\left(t% \right)dt=\begin{cases}\pi i^{n}\mathop{P_{n}\/}\nolimits\!\left(b\right),&-1<% b<1,\\ \frac{1}{2}\pi(\pm i)^{n},&b=\pm 1,\\ 0,&\pm b>1,\end{cases}$

where $\mathop{P_{n}\/}\nolimits$ 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).