DLMF:10.27.E5 (Q3495)

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DLMF:10.27.E5
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    Statements

    test
    K n ⁡ ( z ) = ( - 1 ) n - 1 2 ⁢ ( ∂ ⁡ I ν ⁡ ( z ) ∂ ⁡ ν | ν = n + ∂ ⁡ I ν ⁡ ( z ) ∂ ⁡ ν | ν = - n ) , modified-Bessel-second-kind 𝑛 𝑧 superscript 1 𝑛 1 2 evaluated-at partial-derivative modified-Bessel-first-kind 𝜈 𝑧 𝜈 𝜈 𝑛 evaluated-at partial-derivative modified-Bessel-first-kind 𝜈 𝑧 𝜈 𝜈 𝑛 {\displaystyle{\displaystyle K_{n}\left(z\right)=\frac{(-1)^{n-1}}{2}\*\left(% \left.\frac{\partial I_{\nu}\left(z\right)}{\partial\nu}\right|_{\nu=n}+\left.% \frac{\partial I_{\nu}\left(z\right)}{\partial\nu}\right|_{\nu=-n}\right),}}
    0 references
    DLMF id
    DLMF:10.27.E5
    0 references
    constraint
    n = 0 , ± 1 , ± 2 , … 𝑛 0 plus-or-minus 1 plus-or-minus 2 … {\displaystyle{\displaystyle n=0,\pm 1,\pm 2,\ldots}}
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    Symbols used
    modified Bessel function of the first kind
    test
    I ν ⁡ ( z ) modified-Bessel-first-kind 𝜈 𝑧 {\displaystyle{\displaystyle I_{\NVar{\nu}}\left(\NVar{z}\right)}}
    xml-id
    C10.S25.E2.m2acdec
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    modified Bessel function of the second kind
    test
    K ν ⁡ ( z ) modified-Bessel-second-kind 𝜈 𝑧 {\displaystyle{\displaystyle K_{\NVar{\nu}}\left(\NVar{z}\right)}}
    xml-id
    C10.S25.E3.m2acdec
    0 references
    partial derivative of $$f$$ with respect to $$x$$
    test
    ∂ ⁡ f ∂ ⁡ x partial-derivative 𝑓 𝑥 {\displaystyle{\displaystyle\frac{\partial\NVar{f}}{\partial\NVar{x}}}}
    xml-id
    C1.S5.E3.m4adec
    0 references
    partial differential of $$x$$
    test
    ∂ ⁡ x 𝑥 {\displaystyle{\displaystyle\partial\NVar{x}}}
    xml-id
    C1.S5.E3.m2adec
    0 references
    Q11429
    test
    n 𝑛 {\displaystyle{\displaystyle n}}
    xml-id
    C10.S1.XMD2.m1adec
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