DLMF:10.32.E14 (Q3534)

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DLMF:10.32.E14
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    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:10.32.E14
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    ν - 1 2 , | 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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    Γ ( z ) Euler-Gamma 𝑧 {\displaystyle{\displaystyle\Gamma\left(\NVar{z}\right)}}
    C5.S2.E1.m2addec
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    π {\displaystyle{\displaystyle\pi}}
    C3.S12.E1.m2aldec
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    cos z 𝑧 {\displaystyle{\displaystyle\cos\NVar{z}}}
    C4.S14.E2.m2ahdec
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    d x 𝑥 {\displaystyle{\displaystyle\mathrm{d}\NVar{x}}}
    C1.S4.SS4.m1amdec
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    e {\displaystyle{\displaystyle\mathrm{e}}}
    C4.S2.E11.m2ahdec
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    i imaginary-unit {\displaystyle{\displaystyle\mathrm{i}}}
    C1.S9.E1.m2abdec
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    {\displaystyle{\displaystyle\mathbb{Z}}}
    introduction.Sx4.p2.t1.r20.m2adec
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    {\displaystyle{\displaystyle\int}}
    C1.S4.SS4.m3amdec
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