10 Bessel Functions Kelvin Functions 10.62 Graphs 10.64 Integral Representations

§10.63 Recurrence Relations and Derivatives

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Permalink:: http://dlmf.nist.gov/10.63
See also:: Annotations for Ch.10

§10.63(i) $\operatorname{ber}_{\nu}x$ , $\operatorname{bei}_{\nu}x$ , $\operatorname{ker}_{\nu}x$ , $\operatorname{kei}_{\nu}x$
§10.63(ii) Cross-Products

§10.63(i) $\operatorname{ber}_{\nu}x$ , $\operatorname{bei}_{\nu}x$ , $\operatorname{ker}_{\nu}x$ , $\operatorname{kei}_{\nu}x$

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Keywords:: Kelvin functions, derivatives, recurrence relations
Notes:: Set $z=xe^{3\pi i/4}$ in (10.6.1) and (10.6.2).
Referenced by:: §10.68(ii)
Permalink:: http://dlmf.nist.gov/10.63.i
See also:: Annotations for §10.63 and Ch.10

Let $f_{\nu}(x)$ , $g_{\nu}(x)$ denote any one of the ordered pairs:

10.63.1		$\operatorname{ber}_{\nu}x,\operatorname{bei}_{\nu}x;$
		$\operatorname{bei}_{\nu}x,-\operatorname{ber}_{\nu}x;$
		$\operatorname{ker}_{\nu}x,\operatorname{kei}_{\nu}x;$
		$\operatorname{kei}_{\nu}x,-\operatorname{ker}_{\nu}x.$
ⓘ Symbols: $\operatorname{bei}_{\NVar{\nu}}\left(\NVar{x}\right)$ : Kelvin function, $\operatorname{ber}_{\NVar{\nu}}\left(\NVar{x}\right)$ : Kelvin function, $\operatorname{kei}_{\NVar{\nu}}\left(\NVar{x}\right)$ : Kelvin function, $\operatorname{ker}_{\NVar{\nu}}\left(\NVar{x}\right)$ : Kelvin function, $x$ : real variable and $\nu$ : complex parameter A&S Ref: 9.9.15 Referenced by: §10.71(i) Permalink: http://dlmf.nist.gov/10.63.E1 Encodings: TeX, TeX, TeX, TeX, pMML, pMML, pMML, pMML, png, png, png, png See also: Annotations for §10.63(i), §10.63 and Ch.10

Then

10.63.2	$\displaystyle f_{\nu-1}(x)+f_{\nu+1}(x)$	$\displaystyle=-(\nu\sqrt{2}/x)\left(f_{\nu}(x)-g_{\nu}(x)\right),$
	$\displaystyle f_{\nu+1}(x)+g_{\nu+1}(x)-f_{\nu-1}(x)-g_{\nu-1}(x)=2\sqrt{2}f_{% \nu}^{\prime}(x),$
	$\displaystyle f_{\nu}^{\prime}(x)$	$\displaystyle=-(1/\sqrt{2})\left(f_{\nu-1}(x)+g_{\nu-1}(x)\right)-(\nu/x)f_{% \nu}(x),$
	$\displaystyle f_{\nu}^{\prime}(x)$	$\displaystyle=(1/\sqrt{2})\left(f_{\nu+1}(x)+g_{\nu+1}(x)\right)+(\nu/x)f_{\nu% }(x).$
ⓘ Symbols: $x$ : real variable, $\nu$ : complex parameter, $f_{\nu}$ a Kelvin function and $g_{\nu}$ a Kelvin function A&S Ref: 9.9.14 Referenced by: §10.71(i) Permalink: http://dlmf.nist.gov/10.63.E2 Encodings: TeX, TeX, TeX, TeX, pMML, pMML, pMML, pMML, png, png, png, png See also: Annotations for §10.63(i), §10.63 and Ch.10

10.63.3	$\displaystyle\sqrt{2}\operatorname{ber}'x$	$\displaystyle=\operatorname{ber}_{1}x+\operatorname{bei}_{1}x,$
10.63.3	$\displaystyle\sqrt{2}\operatorname{bei}'x$	$\displaystyle=-\operatorname{ber}_{1}x+\operatorname{bei}_{1}x.$
ⓘ Symbols: $\operatorname{bei}_{\NVar{\nu}}\left(\NVar{x}\right)$ : Kelvin function, $\operatorname{ber}_{\NVar{\nu}}\left(\NVar{x}\right)$ : Kelvin function and $x$ : real variable A&S Ref: 9.9.16 Permalink: http://dlmf.nist.gov/10.63.E3 Encodings: TeX, TeX, pMML, pMML, png, png See also: Annotations for §10.63(i), §10.63 and Ch.10

10.63.4	$\displaystyle\sqrt{2}\operatorname{ker}'x$	$\displaystyle=\operatorname{ker}_{1}x+\operatorname{kei}_{1}x,$
10.63.4	$\displaystyle\sqrt{2}\operatorname{kei}'x$	$\displaystyle=-\operatorname{ker}_{1}x+\operatorname{kei}_{1}x.$
ⓘ Symbols: $\operatorname{kei}_{\NVar{\nu}}\left(\NVar{x}\right)$ : Kelvin function, $\operatorname{ker}_{\NVar{\nu}}\left(\NVar{x}\right)$ : Kelvin function and $x$ : real variable A&S Ref: 9.9.17 Permalink: http://dlmf.nist.gov/10.63.E4 Encodings: TeX, TeX, pMML, pMML, png, png See also: Annotations for §10.63(i), §10.63 and Ch.10

§10.63(ii) Cross-Products

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Keywords:: Kelvin functions, cross-products, recurrence relations
Notes:: Set $a=xe^{3\pi i/4}$ . Then from (10.61.1) and (10.63.5) $J_{\nu}\left(a\right)J_{\nu}\left(\overline{a}\right)=p_{\nu}$ , $J_{\nu}'\left(a\right)J_{\nu}'\left(\overline{a}\right)=s_{\nu}$ , $J_{\nu}\left(a\right)J_{\nu}'\left(\overline{a}\right)=e^{3\pi i/4}(r_{\nu}-iq% _{\nu})$ , $J_{\nu}\left(\overline{a}\right)J_{\nu}'\left(a\right)=e^{-3\pi i/4}(r_{\nu}+% iq_{\nu})$ . Combine these results with (10.6.2) and eliminate the derivatives. See also Petiau (1955, pp. 266–267) (but this reference contains errors). For the functions $\operatorname{ker}_{\nu}x$ and $\operatorname{kei}_{\nu}x$ use the second of (10.61.2).
Referenced by:: Erratum (V1.0.19) for Notation
Permalink:: http://dlmf.nist.gov/10.63.ii
See also:: Annotations for §10.63 and Ch.10

Let

10.63.5	$\displaystyle p_{\nu}$	$\displaystyle={\operatorname{ber}_{\nu}}^{2}x+{\operatorname{bei}_{\nu}}^{2}x,$
	$\displaystyle q_{\nu}$	$\displaystyle=\operatorname{ber}_{\nu}x\operatorname{bei}_{\nu}'x-% \operatorname{ber}_{\nu}'x\operatorname{bei}_{\nu}x,$
	$\displaystyle r_{\nu}$	$\displaystyle=\operatorname{ber}_{\nu}x\operatorname{ber}_{\nu}'x+% \operatorname{bei}_{\nu}x\operatorname{bei}_{\nu}'x,$
	$\displaystyle s_{\nu}$	$\displaystyle=\left(\operatorname{ber}_{\nu}'x\right)^{2}+\left(\operatorname{% bei}_{\nu}'x\right)^{2}.$
ⓘ Defines: $p_{\nu}$ : cross-product (locally), $q_{\nu}$ : cross-product (locally), $r_{\nu}$ : cross-product (locally) and $s_{\nu}$ : cross-product (locally) Symbols: $\operatorname{bei}_{\NVar{\nu}}\left(\NVar{x}\right)$ : Kelvin function, $\operatorname{ber}_{\NVar{\nu}}\left(\NVar{x}\right)$ : Kelvin function, $x$ : real variable and $\nu$ : complex parameter A&S Ref: 9.9.18 Referenced by: §10.63(ii), §10.63(ii) Permalink: http://dlmf.nist.gov/10.63.E5 Encodings: TeX, TeX, TeX, TeX, pMML, pMML, pMML, pMML, png, png, png, png See also: Annotations for §10.63(ii), §10.63 and Ch.10

Then

10.63.6	$\displaystyle p_{\nu+1}$	$\displaystyle=p_{\nu-1}-(4\nu/x)r_{\nu},$
	$\displaystyle q_{\nu+1}$	$\displaystyle=-(\nu/x)p_{\nu}+r_{\nu}=-q_{\nu-1}+2r_{\nu},$
	$\displaystyle r_{\nu+1}$	$\displaystyle=-((\nu+1)/x)p_{\nu+1}+q_{\nu},$
	$\displaystyle s_{\nu}$	$\displaystyle=\tfrac{1}{2}p_{\nu+1}+\tfrac{1}{2}p_{\nu-1}-(\nu^{2}/x^{2})p_{% \nu},$
ⓘ Symbols: $x$ : real variable, $\nu$ : complex parameter, $p_{\nu}$ : cross-product, $q_{\nu}$ : cross-product, $r_{\nu}$ : cross-product and $s_{\nu}$ : cross-product A&S Ref: 9.9.19 Referenced by: §10.63(ii) Permalink: http://dlmf.nist.gov/10.63.E6 Encodings: TeX, TeX, TeX, TeX, pMML, pMML, pMML, pMML, png, png, png, png See also: Annotations for §10.63(ii), §10.63 and Ch.10

and

10.63.7		$p_{\nu}s_{\nu}=r_{\nu}^{2}+q_{\nu}^{2}.$
ⓘ Symbols: $\nu$ : complex parameter, $p_{\nu}$ : cross-product, $q_{\nu}$ : cross-product, $r_{\nu}$ : cross-product and $s_{\nu}$ : cross-product A&S Ref: 9.9.20 Referenced by: §10.63(ii) Permalink: http://dlmf.nist.gov/10.63.E7 Encodings: TeX, pMML, png See also: Annotations for §10.63(ii), §10.63 and Ch.10

Equations (10.63.6) and (10.63.7) also hold when the symbols $\operatorname{ber}$ and $\operatorname{bei}$ in (10.63.5) are replaced throughout by $\operatorname{ker}$ and $\operatorname{kei}$ , respectively.

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§10.63 Recurrence Relations and Derivatives

Contents

§10.63(i) berν⁡x, beiν⁡x, kerν⁡x, keiν⁡x

§10.63(ii) Cross-Products

§10.63(i) $\operatorname{ber}_{\nu}x$ , $\operatorname{bei}_{\nu}x$ , $\operatorname{ker}_{\nu}x$ , $\operatorname{kei}_{\nu}x$