10 Bessel Functions Spherical Bessel Functions 10.51 Recurrence Relations and Derivatives 10.53 Power Series

§10.52 Limiting Forms

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Keywords:: limiting forms, spherical Bessel functions
Referenced by:: §10.47(iii)
Permalink:: http://dlmf.nist.gov/10.52
See also:: Annotations for Ch.10

§10.52(i) $z\to 0$
§10.52(ii) $z\to\infty$

§10.52(i) $z\to 0$

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Notes:: Use §10.53.
Referenced by:: §10.50
Permalink:: http://dlmf.nist.gov/10.52.i
See also:: Annotations for §10.52 and Ch.10

10.52.1 $\mathsf{j}_{n}\left(z\right),{\mathsf{i}^{(1)}_{n}}\left(z\right)\sim z^{n}/(2% n+1)!!,$

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Symbols:: $\sim$ : asymptotic equality, : double factorial, ${\mathsf{i}^{(1)}_{\NVar{n}}}\left(\NVar{z}\right)$ : modified spherical Bessel function, $\mathsf{j}_{\NVar{n}}\left(\NVar{z}\right)$ : spherical Bessel function of the first kind, : integer and : complex variable
A&S Ref:: 10.1.4
Permalink:: http://dlmf.nist.gov/10.52.E1
Encodings:: TeX, pMML, png
See also:: Annotations for §10.52(i), §10.52 and Ch.10

10.52.2 $-\mathsf{y}_{n}\left(z\right),i{\mathsf{h}^{(1)}_{n}}\left(z\right),-i{\mathsf% {h}^{(2)}_{n}}\left(z\right),(-1)^{n}{\mathsf{i}^{(2)}_{n}}\left(z\right),(2/% \pi)\mathsf{k}_{n}\left(z\right)\sim(2n-1)!!/z^{n+1}.$

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Symbols:: $\sim$ : asymptotic equality, $\pi$ : the ratio of the circumference of a circle to its diameter, : double factorial, $\mathrm{i}$ : imaginary unit, ${\mathsf{i}^{(2)}_{\NVar{n}}}\left(\NVar{z}\right)$ : modified spherical Bessel function, $\mathsf{k}_{\NVar{n}}\left(\NVar{z}\right)$ : modified spherical Bessel function, $\mathsf{y}_{\NVar{n}}\left(\NVar{z}\right)$ : spherical Bessel function of the second kind, ${\mathsf{h}^{(1)}_{\NVar{n}}}\left(\NVar{z}\right)$ : spherical Bessel function of the third kind, ${\mathsf{h}^{(2)}_{\NVar{n}}}\left(\NVar{z}\right)$ : spherical Bessel function of the third kind, : integer and : complex variable
A&S Ref:: 10.1.5
Permalink:: http://dlmf.nist.gov/10.52.E2
Encodings:: TeX, pMML, png
See also:: Annotations for §10.52(i), §10.52 and Ch.10

§10.52(ii) $z\to\infty$

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Notes:: Use (10.49.2), (10.49.4), (10.49.6)–(10.49.8), (10.49.10), and (10.49.12).
Referenced by:: §10.50
Permalink:: http://dlmf.nist.gov/10.52.ii
See also:: Annotations for §10.52 and Ch.10

10.52.3

$\mathsf{j}_{n}\left(z\right)=z^{-1}\sin\left(z-\tfrac{1}{2}n\pi\right)+{% \mathrm{e}}^{|\Im z|}O\left(z^{-2}\right),$

$\mathsf{y}_{n}\left(z\right)=-z^{-1}\cos\left(z-\tfrac{1}{2}n\pi\right)+{% \mathrm{e}}^{|\Im z|}O\left(z^{-2}\right),$

10.52.4

${\mathsf{h}^{(1)}_{n}}\left(z\right)\sim i^{-n-1}z^{-1}e^{iz},$

${\mathsf{h}^{(2)}_{n}}\left(z\right)\sim i^{n+1}z^{-1}e^{-iz},$

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Symbols:: $\sim$ : asymptotic equality, $\mathrm{e}$ : base of natural logarithm, $\mathrm{i}$ : imaginary unit, ${\mathsf{h}^{(1)}_{\NVar{n}}}\left(\NVar{z}\right)$ : spherical Bessel function of the third kind, ${\mathsf{h}^{(2)}_{\NVar{n}}}\left(\NVar{z}\right)$ : spherical Bessel function of the third kind, : integer and : complex variable
Permalink:: http://dlmf.nist.gov/10.52.E4
Encodings:: TeX, TeX, pMML, pMML, png, png
See also:: Annotations for §10.52(ii), §10.52 and Ch.10

10.52.5 ${\mathsf{i}^{(1)}_{n}}\left(z\right)\sim{\mathsf{i}^{(2)}_{n}}\left(z\right)% \sim\tfrac{1}{2}z^{-1}e^{z},$ $|\operatorname{ph}z|\leq\tfrac{1}{2}\pi-\delta(<\tfrac{1}{2}\pi)$ ,

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