DLMF:10.43.E21 (Q3635)

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DLMF:10.43.E21
No description defined

    Statements

    Defining formula
    ∫ 0 ∞ sin ⁡ ( a ⁢ t ) ⁢ K 0 ⁡ ( t ) ⁢ d t = arcsinh ⁡ a ( 1 + a 2 ) 1 2 , superscript subscript 0 𝑎 𝑡 modified-Bessel-second-kind 0 𝑡 𝑡 hyperbolic-inverse-sine 𝑎 superscript 1 superscript 𝑎 2 1 2 {\displaystyle{\displaystyle\int_{0}^{\infty}\sin\left(at\right)K_{0}\left(t% \right)\mathrm{d}t=\frac{\operatorname{arcsinh}a}{(1+a^{2})^{\frac{1}{2}}},}}
    0 references
    DLMF id
    DLMF:10.43.E21
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    constraint
    | ℑ ⁡ a | < 1 𝑎 1 {\displaystyle{\displaystyle|\Im a|<1}}
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    Symbols used
    Q10770
    Defining formula
    d x 𝑥 {\displaystyle{\displaystyle\mathrm{d}\NVar{x}}}
    xml-id
    C1.S4.SS4.m1apdec
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    Q11247
    Defining formula
    arcsinh ⁡ z hyperbolic-inverse-sine 𝑧 {\displaystyle{\displaystyle\operatorname{arcsinh}\NVar{z}}}
    xml-id
    C4.S37.SS2.p1.m8adec
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    Q10901
    Defining formula
    ℑ ⁡ absent {\displaystyle{\displaystyle\Im}}
    xml-id
    C1.S9.E2.m2aadec
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    Q10771
    Defining formula
    ∫ {\displaystyle{\displaystyle\int}}
    xml-id
    C1.S4.SS4.m3apdec
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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.m2aidec
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    sine function
    Defining formula
    sin ⁡ z 𝑧 {\displaystyle{\displaystyle\sin\NVar{z}}}
    xml-id
    C4.S14.E1.m2adec
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