DLMF:10.32.E11 (Q3531)

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DLMF:10.32.E11
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    Statements

    test
    K ν ⁡ ( x ⁢ z ) = Γ ⁡ ( ν + 1 2 ) ⁢ ( 2 ⁢ z ) ν π 1 2 ⁢ x ν ⁢ ∫ 0 ∞ cos ⁡ ( x ⁢ t ) ⁢ d t ( t 2 + z 2 ) ν + 1 2 , modified-Bessel-second-kind 𝜈 𝑥 𝑧 Euler-Gamma 𝜈 1 2 superscript 2 𝑧 𝜈 superscript 𝜋 1 2 superscript 𝑥 𝜈 superscript subscript 0 𝑥 𝑡 𝑡 superscript superscript 𝑡 2 superscript 𝑧 2 𝜈 1 2 {\displaystyle{\displaystyle K_{\nu}\left(xz\right)=\frac{\Gamma\left(\nu+% \frac{1}{2}\right)(2z)^{\nu}}{\pi^{\frac{1}{2}}x^{\nu}}\int_{0}^{\infty}\frac{% \cos\left(xt\right)\mathrm{d}t}{(t^{2}+z^{2})^{\nu+\frac{1}{2}}},}}
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    DLMF id
    DLMF:10.32.E11
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    constraint
    ℜ ⁡ ν > - 1 2 𝜈 1 2 {\displaystyle{\displaystyle\Re\nu>-\tfrac{1}{2}}}
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