DLMF:10.32.E14 (Q3534)

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DLMF:10.32.E14
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    Statements

    test
    K ν ⁡ ( z ) = 1 2 ⁢ π 2 ⁢ i ⁢ ( π 2 ⁢ z ) 1 2 ⁢ e - z ⁢ cos ⁡ ( ν ⁢ π ) ⁢ ∫ - i ⁢ ∞ i ⁢ ∞ Γ ⁡ ( t ) ⁢ Γ ⁡ ( 1 2 - t - ν ) ⁢ Γ ⁡ ( 1 2 - t + ν ) ⁢ ( 2 ⁢ z ) t ⁢ d t , modified-Bessel-second-kind 𝜈 𝑧 1 2 superscript 𝜋 2 𝑖 superscript 𝜋 2 𝑧 1 2 superscript 𝑒 𝑧 𝜈 𝜋 superscript subscript 𝑖 𝑖 Euler-Gamma 𝑡 Euler-Gamma 1 2 𝑡 𝜈 Euler-Gamma 1 2 𝑡 𝜈 superscript 2 𝑧 𝑡 𝑡 {\displaystyle{\displaystyle K_{\nu}\left(z\right)=\frac{1}{2\pi^{2}i}\left(% \frac{\pi}{2z}\right)^{\frac{1}{2}}e^{-z}\cos\left(\nu\pi\right)\*\int_{-i% \infty}^{i\infty}\Gamma\left(t\right)\Gamma\left(\tfrac{1}{2}-t-\nu\right)% \Gamma\left(\tfrac{1}{2}-t+\nu\right)(2z)^{t}\mathrm{d}t,}}
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    DLMF id
    DLMF:10.32.E14
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    constraint
    ν - 1 2 ∉ ℤ , | ph ⁡ z | < 3 2 ⁢ π formulae-sequence 𝜈 1 2 phase 𝑧 3 2 𝜋 {\displaystyle{\displaystyle\nu-\tfrac{1}{2}\notin\mathbb{Z},|\operatorname{ph% }z|<\tfrac{3}{2}\pi}}
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    Symbols used
    gamma function
    test
    Γ ⁡ ( z ) Euler-Gamma 𝑧 {\displaystyle{\displaystyle\Gamma\left(\NVar{z}\right)}}
    xml-id
    C5.S2.E1.m2addec
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