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DLMF: §10.52 Limiting Forms ‣ Spherical Bessel Functions ‣ Chapter 10 Bessel Functions

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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

Contents

§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		$\mathsf{j}_{n}\left(z\right),{\mathsf{i}^{(1)}_{n}}\left(z\right)\sim z^{n}/(2% n+1)!!,$
ⓘ 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, $n$ : integer and $z$ : 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		$-\mathsf{y}_{n}\left(z\right),i{\mathsf{h}^{(1)}_{n}}\left(z\right),-i{\mathsf% {h}^{(2)}_{n}}\left(z\right),(-1)^{n}{\mathsf{i}^{(2)}_{n}}\left(z\right),(2/% \pi)\mathsf{k}_{n}\left(z\right)\sim(2n-1)!!/z^{n+1}.$
ⓘ 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, $n$ : integer and $z$ : 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		$\displaystyle\mathsf{j}_{n}\left(z\right)$	$\displaystyle=z^{-1}\sin\left(z-\tfrac{1}{2}n\pi\right)+{\mathrm{e}}^{\|\Im z\|}% O\left(z^{-2}\right),$
10.52.3		$\displaystyle\mathsf{y}_{n}\left(z\right)$	$\displaystyle=-z^{-1}\cos\left(z-\tfrac{1}{2}n\pi\right)+{\mathrm{e}}^{\|\Im z\|% }O\left(z^{-2}\right),$
ⓘ Symbols: $O\left(\NVar{x}\right)$ : order not exceeding, $\pi$ : the ratio of the circumference of a circle to its diameter, $\cos\NVar{z}$ : cosine function, $\mathrm{e}$ : base of natural logarithm, $\Im$ : imaginary part, $\sin\NVar{z}$ : sine function, $\mathsf{j}_{\NVar{n}}\left(\NVar{z}\right)$ : spherical Bessel function of the first kind, $\mathsf{y}_{\NVar{n}}\left(\NVar{z}\right)$ : spherical Bessel function of the second kind, $n$ : integer and $z$ : complex variable Permalink: http://dlmf.nist.gov/10.52.E3 Encodings: TeX, TeX, pMML, pMML, png, png See also: Annotations for §10.52(ii), §10.52 and Ch.10

10.52.4		$\displaystyle{\mathsf{h}^{(1)}_{n}}\left(z\right)$	$\displaystyle\sim i^{-n-1}z^{-1}e^{iz},$
10.52.4		$\displaystyle{\mathsf{h}^{(2)}_{n}}\left(z\right)$	$\displaystyle\sim i^{n+1}z^{-1}e^{-iz},$
ⓘ 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, $n$ : integer and $z$ : 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		${\mathsf{i}^{(1)}_{n}}\left(z\right)\sim{\mathsf{i}^{(2)}_{n}}\left(z\right)% \sim\tfrac{1}{2}z^{-1}e^{z},$
$\|\operatorname{ph}z\|\leq\tfrac{1}{2}\pi-\delta(<\tfrac{1}{2}\pi)$ ,
ⓘ Symbols: $\sim$ : asymptotic equality, $\pi$ : the ratio of the circumference of a circle to its diameter, $\mathrm{e}$ : base of natural logarithm, $\operatorname{ph}$ : phase, ${\mathsf{i}^{(1)}_{\NVar{n}}}\left(\NVar{z}\right)$ : modified spherical Bessel function, ${\mathsf{i}^{(2)}_{\NVar{n}}}\left(\NVar{z}\right)$ : modified spherical Bessel function, $n$ : integer, $z$ : complex variable and $\delta$ : small positive constant Permalink: http://dlmf.nist.gov/10.52.E5 Encodings: TeX, pMML, png See also: Annotations for §10.52(ii), §10.52 and Ch.10

10.52.6		$\mathsf{k}_{n}\left(z\right)\sim\tfrac{1}{2}\pi z^{-1}e^{-z}.$
ⓘ Symbols: $\sim$ : asymptotic equality, $\pi$ : the ratio of the circumference of a circle to its diameter, $\mathrm{e}$ : base of natural logarithm, $\mathsf{k}_{\NVar{n}}\left(\NVar{z}\right)$ : modified spherical Bessel function, $n$ : integer and $z$ : complex variable Permalink: http://dlmf.nist.gov/10.52.E6 Encodings: TeX, pMML, png See also: Annotations for §10.52(ii), §10.52 and Ch.10

10.51 Recurrence Relations and Derivatives 10.53 Power Series

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