10 Bessel Functions Spherical Bessel Functions 10.56 Generating Functions 10.58 Zeros

§10.57 Uniform Asymptotic Expansions for Large Order

ⓘ

Keywords:: asymptotic approximations for large order, spherical Bessel functions, uniform asymptotic expansions for large order
Notes:: For (10.57.1) use the differentiated form of the first of (10.47.3).
Permalink:: http://dlmf.nist.gov/10.57
See also:: Annotations for Ch.10

Asymptotic expansions for $\mathsf{j}_{n}\left((n+\tfrac{1}{2})z\right)$ , $\mathsf{y}_{n}\left((n+\tfrac{1}{2})z\right)$ , ${\mathsf{h}^{(1)}_{n}}\left((n+\tfrac{1}{2})z\right)$ , ${\mathsf{h}^{(2)}_{n}}\left((n+\tfrac{1}{2})z\right)$ , ${\mathsf{i}^{(1)}_{n}}\left((n+\tfrac{1}{2})z\right)$ , and $\mathsf{k}_{n}\left((n+\tfrac{1}{2})z\right)$ as $n\to\infty$ that are uniform with respect to $z$ can be obtained from the results given in §§10.20 and 10.41 by use of the definitions (10.47.3)–(10.47.7) and (10.47.9). Subsequently, for ${\mathsf{i}^{(2)}_{n}}\left((n+\tfrac{1}{2})z\right)$ the connection formula (10.47.11) is available.

For the corresponding expansion for $\mathsf{j}_{n}'\left((n+\tfrac{1}{2})z\right)$ use

10.57.1		$\mathsf{j}_{n}'\left((n+\tfrac{1}{2})z\right)=\frac{\pi^{\frac{1}{2}}}{((2n+1)% z)^{\frac{1}{2}}}J_{n+\frac{1}{2}}'\left((n+\tfrac{1}{2})z\right)-\frac{\pi^{% \frac{1}{2}}}{((2n+1)z)^{\frac{3}{2}}}J_{n+\frac{1}{2}}\left((n+\tfrac{1}{2})z% \right).$
ⓘ Symbols: $J_{\NVar{\nu}}\left(\NVar{z}\right)$ : Bessel function of the first kind, $\pi$ : the ratio of the circumference of a circle to its diameter, $\mathsf{j}_{\NVar{n}}\left(\NVar{z}\right)$ : spherical Bessel function of the first kind, $n$ : integer and $z$ : complex variable Referenced by: §10.57 Permalink: http://dlmf.nist.gov/10.57.E1 Encodings: TeX, pMML, png See also: Annotations for §10.57 and Ch.10

Similarly for the expansions of the derivatives of the other six functions.

10.56 Generating Functions 10.58 Zeros

Site Privacy Accessibility Privacy Program Copyrights Vulnerability Disclosure No Fear Act Policy FOIA Environmental Policy Scientific Integrity Information Quality Standards Commerce.gov Science.gov USA.gov