DLMF:10.32.E6 (Q3526)

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DLMF:10.32.E6
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    Statements

    Defining formula
    K 0 ⁡ ( x ) = ∫ 0 ∞ cos ⁡ ( x ⁢ sinh ⁡ t ) ⁢ d t = ∫ 0 ∞ cos ⁡ ( x ⁢ t ) t 2 + 1 ⁢ d t , modified-Bessel-second-kind 0 𝑥 superscript subscript 0 𝑥 𝑡 𝑡 superscript subscript 0 𝑥 𝑡 superscript 𝑡 2 1 𝑡 {\displaystyle{\displaystyle K_{0}\left(x\right)=\int_{0}^{\infty}\cos\left(x% \sinh t\right)\mathrm{d}t=\int_{0}^{\infty}\frac{\cos\left(xt\right)}{\sqrt{t^% {2}+1}}\mathrm{d}t,}}
    0 references
    DLMF id
    DLMF:10.32.E6
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    constraint
    x > 0 𝑥 0 {\displaystyle{\displaystyle x>0}}
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    Symbols used
    cosine function
    Defining formula
    cos ⁡ z 𝑧 {\displaystyle{\displaystyle\cos\NVar{z}}}
    xml-id
    C4.S14.E2.m2aedec
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    Q10770
    Defining formula
    d x 𝑥 {\displaystyle{\displaystyle\mathrm{d}\NVar{x}}}
    xml-id
    C1.S4.SS4.m1aedec
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    hyperbolic sine function
    Defining formula
    sinh ⁡ z 𝑧 {\displaystyle{\displaystyle\sinh\NVar{z}}}
    xml-id
    C4.S28.E1.m2adec
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    Q10771
    Defining formula
    ∫ {\displaystyle{\displaystyle\int}}
    xml-id
    C1.S4.SS4.m3aedec
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    modified Bessel function of the second kind
    Defining formula
    K ν ⁡ ( z ) modified-Bessel-second-kind 𝜈 𝑧 {\displaystyle{\displaystyle K_{\NVar{\nu}}\left(\NVar{z}\right)}}
    xml-id
    C10.S25.E3.m2aadec
    0 references
    Q11435
    Defining formula
    x 𝑥 {\displaystyle{\displaystyle x}}
    xml-id
    C10.S1.XMD4.m1dec
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