10 Bessel FunctionsKelvin Functions10.62 Graphs10.64 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

Contents

1. §10.63(i) ${\mathrm{ber}}_{\nu }x$, ${\mathrm{bei}}_{\nu }x$, ${\ker }_{\nu }x$, ${\mathrm{kei}}_{\nu }x$
2. §10.63(ii) Cross-Products

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

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Keywords:

Kelvin functions, derivatives, recurrence relations

Notes:

Set $z=x{\mathrm{e}}^{3\pi \mathrm{i}/4}$ 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		${\mathrm{ber}}_{\nu }x,{\mathrm{bei}}_{\nu }x;$
		${\mathrm{bei}}_{\nu }x,-{\mathrm{ber}}_{\nu }x;$
		${\ker }_{\nu }x,{\mathrm{kei}}_{\nu }x;$
		${\mathrm{kei}}_{\nu }x,-{\ker }_{\nu }x.$
ⓘ Symbols: ${\mathrm{bei}}_{\nu }\left(x\right)$: Kelvin function, ${\mathrm{ber}}_{\nu }\left(x\right)$: Kelvin function, ${\mathrm{kei}}_{\nu }\left(x\right)$: Kelvin function, ${\ker }_{\nu }\left(x\right)$: 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		$f_{\nu -1}(x)+f_{\nu +1}(x)$	$=-(\nu \sqrt{2}/x)\left(f_{\nu }(x)-g_{\nu }(x)\right),$
		$f_{\nu +1}(x)+g_{\nu +1}(x)-f_{\nu -1}(x)-g_{\nu -1}(x)=2\sqrt{2}{f}_{\nu }^{\prime }(x),$
		$f_{\nu }^{\prime }(x)$	$=-(1/\sqrt{2})\left(f_{\nu -1}(x)+g_{\nu -1}(x)\right)-(\nu /x)f_{\nu }(x),$
		$f_{\nu }^{\prime }(x)$	$=(1/\sqrt{2})\left(f_{\nu +1}(x)+g_{\nu +1}(x)\right)+(\nu /x)f_{\nu }(x).$
ⓘ 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		$\sqrt{2}{\mathrm{ber}}^{\prime }x$	$={\mathrm{ber}}_{1}x+{\mathrm{bei}}_{1}x,$
		$\sqrt{2}{\mathrm{bei}}^{\prime }x$	$=-{\mathrm{ber}}_{1}x+{\mathrm{bei}}_{1}x.$
ⓘ Symbols: ${\mathrm{bei}}_{\nu }\left(x\right)$: Kelvin function, ${\mathrm{ber}}_{\nu }\left(x\right)$: 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		$\sqrt{2}{\ker }^{\prime }x$	$={\ker }_{1}x+{\mathrm{kei}}_{1}x,$
		$\sqrt{2}{\mathrm{kei}}^{\prime }x$	$=-{\ker }_{1}x+{\mathrm{kei}}_{1}x.$
ⓘ Symbols: ${\mathrm{kei}}_{\nu }\left(x\right)$: Kelvin function, ${\ker }_{\nu }\left(x\right)$: 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 $a=x{\mathrm{e}}^{3\pi \mathrm{i}/4}$. Then from (10.61.1) and (10.63.5) $J_{\nu }\left(a\right)J_{\nu }\left(\overline{a}\right)=p_{\nu }$, $J_{\nu }^{\prime }\left(a\right)J_{\nu }^{\prime }\left(\overline{a}\right)=s_{\nu }$, $J_{\nu }\left(a\right)J_{\nu }^{\prime }\left(\overline{a}\right)={\mathrm{e}}^{3\pi \mathrm{i}/4}(r_{\nu }-\mathrm{i}{q}_{\nu })$, $J_{\nu }\left(\overline{a}\right)J_{\nu }^{\prime }\left(a\right)={\mathrm{e}}^{-3\pi \mathrm{i}/4}(r_{\nu }+\mathrm{i}{q}_{\nu })$. 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 ${\ker }_{\nu }x$ and ${\mathrm{kei}}_{\nu }x$ 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		$p_{\nu }$	$={\mathrm{ber}}_{\nu }^{2}x+{\mathrm{bei}}_{\nu }^{2}x,$
		$q_{\nu }$	$={\mathrm{ber}}_{\nu }x{\mathrm{bei}}_{\nu }^{\prime }x-{\mathrm{ber}}_{\nu }^{\prime }x{\mathrm{bei}}_{\nu }x,$
		$r_{\nu }$	$={\mathrm{ber}}_{\nu }x{\mathrm{ber}}_{\nu }^{\prime }x+{\mathrm{bei}}_{\nu }x{\mathrm{bei}}_{\nu }^{\prime }x,$
		$s_{\nu }$	$={\left({\mathrm{ber}}_{\nu }^{\prime }x\right)}^2+{\left({\mathrm{bei}}_{\nu }^{\prime }x\right)}^2.$
ⓘ Defines: $p_{\nu }$: cross-product (locally), $q_{\nu }$: cross-product (locally), $r_{\nu }$: cross-product (locally) and $s_{\nu }$: cross-product (locally) Symbols: ${\mathrm{bei}}_{\nu }\left(x\right)$: Kelvin function, ${\mathrm{ber}}_{\nu }\left(x\right)$: 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		$p_{\nu +1}$	$=p_{\nu -1}-(4\nu /x)r_{\nu },$
		$q_{\nu +1}$	$=-(\nu /x)p_{\nu }+r_{\nu }=-q_{\nu -1}+2r_{\nu },$
		$r_{\nu +1}$	$=-((\nu +1)/x)p_{\nu +1}+q_{\nu },$
		$s_{\nu }$	$=\frac{1}{2}{p}_{\nu +1}+\frac{1}{2}{p}_{\nu -1}-({\nu }^2/x^2)p_{\nu },$
ⓘ 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 $\mathrm{ber}$ and $\mathrm{bei}$ in (10.63.5) are replaced throughout by $\ker$ and $\mathrm{kei}$, respectively.

10.62 Graphs10.64 Integral Representations

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