DLMF:10.61.E4 (Q3793)

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DLMF:10.61.E4
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    Statements

    Defining formula
    x 4 ⁢ d 4 w d x 4 + 2 ⁢ x 3 ⁢ d 3 w d x 3 - ( 1 + 2 ⁢ ν 2 ) ⁢ ( x 2 ⁢ d 2 w d x 2 - x ⁢ d w d x ) + ( ν 4 - 4 ⁢ ν 2 + x 4 ) ⁢ w = 0 , superscript 𝑥 4 derivative 𝑤 𝑥 4 2 superscript 𝑥 3 derivative 𝑤 𝑥 3 1 2 superscript 𝜈 2 superscript 𝑥 2 derivative 𝑤 𝑥 2 𝑥 derivative 𝑤 𝑥 superscript 𝜈 4 4 superscript 𝜈 2 superscript 𝑥 4 𝑤 0 {\displaystyle{\displaystyle x^{4}\frac{{\mathrm{d}}^{4}w}{{\mathrm{d}x}^{4}}+% 2x^{3}\frac{{\mathrm{d}}^{3}w}{{\mathrm{d}x}^{3}}-(1+2\nu^{2})\left(x^{2}\frac% {{\mathrm{d}}^{2}w}{{\mathrm{d}x}^{2}}-x\frac{\mathrm{d}w}{\mathrm{d}x}\right)% +(\nu^{4}-4\nu^{2}+x^{4})w=0,}}
    0 references
    DLMF id
    DLMF:10.61.E4
    0 references
    constraint
    w = ber ± ν ⁡ x , bei ± ν ⁡ x , ker ± ν ⁡ x , kei ± ν ⁡ x 𝑤 Kelvin-ber plus-or-minus 𝜈 𝑥 Kelvin-bei plus-or-minus 𝜈 𝑥 Kelvin-ker plus-or-minus 𝜈 𝑥 Kelvin-kei plus-or-minus 𝜈 𝑥 {\displaystyle{\displaystyle w=\operatorname{ber}_{\pm\nu}x,\operatorname{bei}% _{\pm\nu}x,\operatorname{ker}_{\pm\nu}x,\operatorname{kei}_{\pm\nu}x}}
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    Symbols used
    Kelvin function
    Defining formula
    bei ν ⁡ ( x ) Kelvin-bei 𝜈 𝑥 {\displaystyle{\displaystyle\operatorname{bei}_{\NVar{\nu}}\left(\NVar{x}% \right)}}
    xml-id
    C10.S61.E1.m3abdec
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    Kelvin function
    Defining formula
    ber ν ⁡ ( x ) Kelvin-ber 𝜈 𝑥 {\displaystyle{\displaystyle\operatorname{ber}_{\NVar{\nu}}\left(\NVar{x}% \right)}}
    xml-id
    C10.S61.E1.m2abdec
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    Kelvin function
    Defining formula
    kei ν ⁡ ( x ) Kelvin-kei 𝜈 𝑥 {\displaystyle{\displaystyle\operatorname{kei}_{\NVar{\nu}}\left(\NVar{x}% \right)}}
    xml-id
    C10.S61.E2.m3abdec
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    Kelvin function
    Defining formula
    ker ν ⁡ ( x ) Kelvin-ker 𝜈 𝑥 {\displaystyle{\displaystyle\operatorname{ker}_{\NVar{\nu}}\left(\NVar{x}% \right)}}
    xml-id
    C10.S61.E2.m2abdec
    0 references
    derivative of $$f$$ with respect to $$x$$
    Defining formula
    d f d x derivative 𝑓 𝑥 {\displaystyle{\displaystyle\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}}}
    xml-id
    C1.S4.E4.m2aadec
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    Q11435
    Defining formula
    x 𝑥 {\displaystyle{\displaystyle x}}
    xml-id
    C10.S1.XMD4.m1cdec
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    Q11427
    Defining formula
    ν 𝜈 {\displaystyle{\displaystyle\nu}}
    xml-id
    C10.S1.XMD7.m1cdec
    0 references
     
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