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

§10.51 Recurrence Relations and Derivatives

ⓘ

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

§10.51(i) Unmodified Functions

ⓘ

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

ⓘ

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

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

Note: This service is not intended for secure transactions such as banking, social media, email, or purchasing. Use at your own risk. We assume no liability whatsoever for broken pages.

§10.51 Recurrence Relations and Derivatives

Contents

§10.51(i) Unmodified Functions

§10.51(ii) Modified Functions

Pfad - The Proxy pFad of © 2024 Garber Painting. All rights reserved.