10 Bessel Functions Spherical Bessel Functions 10.58 Zeros 10.60 Sums

§10.59 Integrals

ⓘ

Keywords:: integrals, spherical Bessel functions
Notes:: For (10.59.1) suppose first $b\neq 0$ . The left-hand side is $2i\int_{0}^{\infty}\sin\left(bt\right)\mathsf{j}_{n}\left(t\right)\,\mathrm{d}t$ or $2\int_{0}^{\infty}\cos\left(bt\right)\mathsf{j}_{n}\left(t\right)\,\mathrm{d}t$ according as $n$ is odd or even, see (10.47.14). Next, apply (10.22.64) with $a=1$ , $\mu=\tfrac{1}{2}$ or $-\tfrac{1}{2}$ , and subsequently replace $2n+1$ or $2n$ by $n$ . For $J_{\pm(\ifrac{1}{2})}\left(bt\right)$ and $J_{n+(\ifrac{1}{2})}\left(t\right)$ 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 $P_{n}\left(0\right)=(-1)^{\frac{1}{2}n}\ifrac{{\left(\frac{1}{2}\right)_{\frac% {1}{2}n}}}{(\frac{1}{2}n)!}$ or $0$ , according as the nonnegative integer $n$ is even or odd; see (14.5.1) and §5.5.
Referenced by:: Erratum (V1.1.6) for Chapters 10 Bessel Functions, 18 Orthogonal Polynomials, 34 3j, 6j, 9j Symbols
Permalink:: http://dlmf.nist.gov/10.59
See also:: Annotations for Ch.10

10.59.1		$\int_{-\infty}^{\infty}e^{ibt}\mathsf{j}_{n}\left(t\right)\,\mathrm{d}t=\begin% {cases}\pi i^{n}P_{n}\left(b\right),&-1<b<1,\\ \frac{1}{2}\pi(\pm\mathrm{i})^{n},&b=\pm 1,\\ 0,&\pm b>1,\end{cases}$
ⓘ Symbols: $P_{\NVar{n}}\left(\NVar{x}\right)$ : Legendre polynomial, $\pi$ : the ratio of the circumference of a circle to its diameter, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\mathrm{e}$ : base of natural logarithm, $\mathrm{i}$ : imaginary unit, $\int$ : integral, $\mathsf{j}_{\NVar{n}}\left(\NVar{z}\right)$ : spherical Bessel function of the first kind and $n$ : integer A&S Ref: 11.4.26 Referenced by: §10.59 Permalink: http://dlmf.nist.gov/10.59.E1 Encodings: TeX, pMML, png See also: Annotations for §10.59 and Ch.10

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).

10.58 Zeros 10.60 Sums

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§10.59 Integrals

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