10 Bessel Functions Spherical Bessel Functions 10.50 Wronskians and Cross-Products 10.52 Limiting Forms

§10.51 Recurrence Relations and Derivatives

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Permalink:: http://dlmf.nist.gov/10.51
See also:: Annotations for Ch.10

§10.51(i) Unmodified Functions

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Keywords:: derivatives, recurrence relations, spherical Bessel functions
Notes:: For (10.51.1) and (10.51.2) combine (10.6.1) and (10.6.2) with the definitions (10.47.3)–(10.47.5). For (10.51.3) apply induction with the aid of (10.51.2).
Permalink:: http://dlmf.nist.gov/10.51.i
See also:: Annotations for §10.51 and Ch.10

Let $f_{n}(z)$ denote any of $\mathsf{j}_{n}\left(z\right)$ , $\mathsf{y}_{n}\left(z\right)$ , ${\mathsf{h}^{(1)}_{n}}\left(z\right)$ , or ${\mathsf{h}^{(2)}_{n}}\left(z\right)$ . Then

10.51.1		$\displaystyle f_{n-1}(z)+f_{n+1}(z)$	$\displaystyle=((2n+1)/z)f_{n}(z),$
		$\displaystyle nf_{n-1}(z)-(n+1)f_{n+1}(z)$	$\displaystyle=(2n+1)f_{n}^{\prime}(z),$
	$n=1,2,\dots$ ,
	ⓘ Symbols: $n$ : integer, $z$ : complex variable and $f_{n}(z)$ : a spherical Bessel function A&S Ref: 10.1.19, 10.1.20 Referenced by: §10.50, §10.51(i), §10.56 Permalink: http://dlmf.nist.gov/10.51.E1 Encodings: TeX, TeX, pMML, pMML, png, png See also: Annotations for §10.51(i), §10.51 and Ch.10

10.51.2		$\displaystyle f_{n}^{\prime}(z)$	$\displaystyle=f_{n-1}(z)-((n+1)/z)f_{n}(z),$
	$n=1,2,\dots$ ,
		$\displaystyle f_{n}^{\prime}(z)$	$\displaystyle=-f_{n+1}(z)+(n/z)f_{n}(z),$
	$n=0,1,\dots.$
ⓘ Symbols: $n$ : integer, $z$ : complex variable and $f_{n}(z)$ : a spherical Bessel function A&S Ref: 10.1.21, 10.1.22 Referenced by: §10.50, §10.51(i) Permalink: http://dlmf.nist.gov/10.51.E2 Encodings: TeX, TeX, pMML, pMML, png, png See also: Annotations for §10.51(i), §10.51 and Ch.10

10.51.3		$\displaystyle\left(\frac{1}{z}\frac{\mathrm{d}}{\mathrm{d}z}\right)^{m}(z^{n+1% }f_{n}(z))$	$\displaystyle=z^{n-m+1}f_{n-m}(z),$
	$m=0,1,\dots,n$ ,
		$\displaystyle\left(\frac{1}{z}\frac{\mathrm{d}}{\mathrm{d}z}\right)^{m}(z^{-n}% f_{n}(z))$	$\displaystyle=(-1)^{m}z^{-n-m}f_{n+m}(z),$
	$m=0,1,\dots.$
ⓘ Symbols: $\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}$ : derivative of $f$ with respect to $x$ , $m$ : integer, $n$ : integer, $z$ : complex variable and $f_{n}(z)$ : a spherical Bessel function A&S Ref: 10.1.23, 10.1.24 Referenced by: §10.49(iii), §10.51(i) Permalink: http://dlmf.nist.gov/10.51.E3 Encodings: TeX, TeX, pMML, pMML, png, png See also: Annotations for §10.51(i), §10.51 and Ch.10

§10.51(ii) Modified Functions

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Keywords:: derivatives, recurrence relations, spherical Bessel functions
Notes:: For (10.51.4) and (10.51.5) combine (10.29.1) and (10.29.2) with the definitions (10.47.7) and (10.47.9). For (10.51.6) apply induction with the aid of (10.51.5).
Permalink:: http://dlmf.nist.gov/10.51.ii
See also:: Annotations for §10.51 and Ch.10

Let $g_{n}(z)$ denote ${\mathsf{i}^{(1)}_{n}}\left(z\right)$ , ${\mathsf{i}^{(2)}_{n}}\left(z\right)$ , or $(-1)^{n}$ $\mathsf{k}_{n}\left(z\right)$ . Then

10.51.4		$\displaystyle g_{n-1}(z)-g_{n+1}(z)$	$\displaystyle=((2n+1)/z)g_{n}(z)$
		$\displaystyle ng_{n-1}(z)+(n+1)g_{n+1}(z)$	$\displaystyle=(2n+1)g_{n}^{\prime}(z),$
	$n=1,2,\dotsc$ ,
	ⓘ Symbols: $n$ : integer, $z$ : complex variable and $g_{n}(z)$ : a spherical Bessel function A&S Ref: 10.2.18, 10.2.19 Referenced by: §10.51(ii) Permalink: http://dlmf.nist.gov/10.51.E4 Encodings: TeX, TeX, pMML, pMML, png, png See also: Annotations for §10.51(ii), §10.51 and Ch.10

10.51.5		$\displaystyle g_{n}^{\prime}(z)$	$\displaystyle=g_{n-1}(z)-((n+1)/z)g_{n}(z),$
	$n=1,2,\dotsc$ ,
		$\displaystyle g_{n}^{\prime}(z)$	$\displaystyle=g_{n+1}(z)+(n/z)g_{n}(z),$
	$n=0,1,\dotsc.$
ⓘ Symbols: $n$ : integer, $z$ : complex variable and $g_{n}(z)$ : a spherical Bessel function A&S Ref: 10.2.20, 10.2.21 Referenced by: §10.51(ii) Permalink: http://dlmf.nist.gov/10.51.E5 Encodings: TeX, TeX, pMML, pMML, png, png See also: Annotations for §10.51(ii), §10.51 and Ch.10

10.51.6		$\displaystyle\left(\frac{1}{z}\frac{\mathrm{d}}{\mathrm{d}z}\right)^{m}(z^{n+1% }g_{n}(z))$	$\displaystyle=z^{n-m+1}g_{n-m}(z),$
	$m=0,1,\dotsc,n$ ,
		$\displaystyle\left(\frac{1}{z}\frac{\mathrm{d}}{\mathrm{d}z}\right)^{m}(z^{-n}% g_{n}(z))$	$\displaystyle=z^{-n-m}g_{n+m}(z),$
	$m=0,1,\dotsc.$
ⓘ Symbols: $\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}$ : derivative of $f$ with respect to $x$ , $m$ : integer, $n$ : integer, $z$ : complex variable and $g_{n}(z)$ : a spherical Bessel function A&S Ref: 10.2.22, 10.2.23 Referenced by: §10.51(ii) Permalink: http://dlmf.nist.gov/10.51.E6 Encodings: TeX, TeX, pMML, pMML, png, png See also: Annotations for §10.51(ii), §10.51 and Ch.10

10.50 Wronskians and Cross-Products 10.52 Limiting Forms

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§10.51 Recurrence Relations and Derivatives

Contents

§10.51(i) Unmodified Functions

§10.51(ii) Modified Functions

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