§10.63 Recurrence Relations and Derivatives

Permalink:: http://dlmf.nist.gov/10.63

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

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

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

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

10.63.1

$\mathop{\mathrm{ber}_{{\nu}}\/}\nolimits x,\mathop{\mathrm{bei}_{{\nu}}\/}\nolimits x;$

$\mathop{\mathrm{bei}_{{\nu}}\/}\nolimits x,-\mathop{\mathrm{ber}_{{\nu}}\/}\nolimits x;$

$\mathop{\mathrm{ker}_{{\nu}}\/}\nolimits x,\mathop{\mathrm{kei}_{{\nu}}\/}\nolimits x;$

$\mathop{\mathrm{kei}_{{\nu}}\/}\nolimits x,-\mathop{\mathrm{ker}_{{\nu}}\/}\nolimits x.$

Symbols:: $\mathop{\mathrm{bei}_{{\nu}}\/}\nolimits\!\left(x\right)$ : Kelvin function, $\mathop{\mathrm{ber}_{{\nu}}\/}\nolimits\!\left(x\right)$ : Kelvin function, $\mathop{\mathrm{kei}_{{\nu}}\/}\nolimits\!\left(x\right)$ : Kelvin function, $\mathop{\mathrm{ker}_{{\nu}}\/}\nolimits\!\left(x\right)$ : 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
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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
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10.63.3

$\sqrt{2}{\mathop{\mathrm{ber}\/}\nolimits^{{\prime}}}x=\mathop{\mathrm{ber}_{{1}}\/}\nolimits x+\mathop{\mathrm{bei}_{{1}}\/}\nolimits x,$

$\sqrt{2}{\mathop{\mathrm{bei}\/}\nolimits^{{\prime}}}x=-\mathop{\mathrm{ber}_{{1}}\/}\nolimits x+\mathop{\mathrm{bei}_{{1}}\/}\nolimits x.$

Symbols:: $\mathop{\mathrm{bei}_{{\nu}}\/}\nolimits\!\left(x\right)$ : Kelvin function, $\mathop{\mathrm{ber}_{{\nu}}\/}\nolimits\!\left(x\right)$ : Kelvin function and $x$ : real variable
A&S Ref:: 9.9.16
Permalink:: http://dlmf.nist.gov/10.63.E3
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10.63.4

$\sqrt{2}{\mathop{\mathrm{ker}\/}\nolimits^{{\prime}}}x=\mathop{\mathrm{ker}_{{1}}\/}\nolimits x+\mathop{\mathrm{kei}_{{1}}\/}\nolimits x,$

$\sqrt{2}{\mathop{\mathrm{kei}\/}\nolimits^{{\prime}}}x=-\mathop{\mathrm{ker}_{{1}}\/}\nolimits x+\mathop{\mathrm{kei}_{{1}}\/}\nolimits x.$

Symbols:: $\mathop{\mathrm{kei}_{{\nu}}\/}\nolimits\!\left(x\right)$ : Kelvin function, $\mathop{\mathrm{ker}_{{\nu}}\/}\nolimits\!\left(x\right)$ : Kelvin function and $x$ : real variable
A&S Ref:: 9.9.17
Permalink:: http://dlmf.nist.gov/10.63.E4
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§10.63(ii) Cross-Products

Notes:: Set $a=xe^{{3\pi i/4}}$ . Then from (10.61.1) and (10.63.5) $\mathop{J_{{\nu}}\/}\nolimits\!\left(a\right)\mathop{J_{{\nu}}\/}\nolimits\!\left(\bar{a}\right)=p_{\nu}$ , ${\mathop{J_{{\nu}}\/}\nolimits^{{\prime}}}\!\left(a\right){\mathop{J_{{\nu}}\/}\nolimits^{{\prime}}}\!\left(\bar{a}\right)=s_{\nu}$ , $\mathop{J_{{\nu}}\/}\nolimits\!\left(a\right){\mathop{J_{{\nu}}\/}\nolimits^{{\prime}}}\!\left(\bar{a}\right)=e^{{3\pi i/4}}(r_{\nu}-iq_{\nu})$ , $\mathop{J_{{\nu}}\/}\nolimits\!\left(\bar{a}\right){\mathop{J_{{\nu}}\/}\nolimits^{{\prime}}}\!\left(a\right)=e^{{-3\pi i/4}}(r_{\nu}+iq_{\nu})$ . 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 $\mathop{\mathrm{ker}_{{\nu}}\/}\nolimits x$ and $\mathop{\mathrm{kei}_{{\nu}}\/}\nolimits x$ use the second of (10.61.2).
Keywords:: Kelvin functions
Permalink:: http://dlmf.nist.gov/10.63.ii

Let

10.63.5

$p_{\nu}={\mathop{\mathrm{ber}_{{\nu}}\/}\nolimits^{{2}}}x+{\mathop{\mathrm{bei}_{{\nu}}\/}\nolimits^{{2}}}x,$

$q_{\nu}=\mathop{\mathrm{ber}_{{\nu}}\/}\nolimits x{\mathop{\mathrm{bei}_{{\nu}}\/}\nolimits^{{\prime}}}x-{\mathop{\mathrm{ber}_{{\nu}}\/}\nolimits^{{\prime}}}x\mathop{\mathrm{bei}_{{\nu}}\/}\nolimits x,$

$r_{\nu}=\mathop{\mathrm{ber}_{{\nu}}\/}\nolimits x{\mathop{\mathrm{ber}_{{\nu}}\/}\nolimits^{{\prime}}}x+\mathop{\mathrm{bei}_{{\nu}}\/}\nolimits x{\mathop{\mathrm{bei}_{{\nu}}\/}\nolimits^{{\prime}}}x,$

$s_{\nu}=\left({\mathop{\mathrm{ber}_{{\nu}}\/}\nolimits^{{\prime}}}x\right)^{2}+\left({\mathop{\mathrm{bei}_{{\nu}}\/}\nolimits^{{\prime}}}x\right)^{2}.$

Symbols:: $\mathop{\mathrm{bei}_{{\nu}}\/}\nolimits\!\left(x\right)$ : Kelvin function, $\mathop{\mathrm{ber}_{{\nu}}\/}\nolimits\!\left(x\right)$ : Kelvin function, $x$ : 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:: 9.9.18
Referenced by:: §10.63(ii), §10.63(ii)
Permalink:: http://dlmf.nist.gov/10.63.E5
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Then

10.63.6

$p_{{\nu+1}}=p_{{\nu-1}}-(4\nu/x)r_{\nu},$

$q_{{\nu+1}}=-(\nu/x)p_{\nu}+r_{\nu}=-q_{{\nu-1}}+2r_{\nu},$

$r_{{\nu+1}}=-((\nu+1)/x)p_{{\nu+1}}+q_{\nu},$

$s_{\nu}=\tfrac{1}{2}p_{{\nu+1}}+\tfrac{1}{2}p_{{\nu-1}}-(\nu^{2}/x^{2})p_{\nu},$

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
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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

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

§10.63 Recurrence Relations and Derivatives

Contents

§10.63(i) , , ,

§10.63(ii) Cross-Products

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