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10 Bessel FunctionsKelvin Functions

§10.63 Recurrence Relations and Derivatives

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§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

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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:

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).
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.
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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
Encodings:
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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.
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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
Encodings:
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§10.63(ii) Cross-Products

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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}.
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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
Encodings:
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Then

and

10.63.7p_{\nu}s_{\nu}=r_{\nu}^{2}+q_{\nu}^{2}.

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.

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