DLMF:10.22.E40 (Q3414)

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DLMF:10.22.E40
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    Statements

    test
    ∫ x ∞ Y 0 ⁡ ( t ) t ⁢ d t = - 1 π ⁢ ( ln ⁡ ( 1 2 ⁢ x ) + γ ) 2 + π 6 + 2 π ⁢ ∑ k = 1 ∞ ( - 1 ) k ⁢ ( ψ ⁡ ( k + 1 ) + 1 2 ⁢ k - ln ⁡ ( 1 2 ⁢ x ) ) ⁢ ( 1 2 ⁢ x ) 2 ⁢ k 2 ⁢ k ⁢ ( k ! ) 2 , superscript subscript 𝑥 Bessel-Y-Weber 0 𝑡 𝑡 𝑡 1 𝜋 superscript 1 2 𝑥 2 𝜋 6 2 𝜋 superscript subscript 𝑘 1 superscript 1 𝑘 digamma 𝑘 1 1 2 𝑘 1 2 𝑥 superscript 1 2 𝑥 2 𝑘 2 𝑘 superscript 𝑘 2 {\displaystyle{\displaystyle\int_{x}^{\infty}\frac{Y_{0}\left(t\right)}{t}% \mathrm{d}t=-\frac{1}{\pi}\left(\ln\left(\tfrac{1}{2}x\right)+\gamma\right)^{2% }+\frac{\pi}{6}+\frac{2}{\pi}\sum_{k=1}^{\infty}(-1)^{k}\*\left(\psi\left(k+1% \right)+\frac{1}{2k}-\ln\left(\tfrac{1}{2}x\right)\right)\frac{(\tfrac{1}{2}x)% ^{2k}}{2k(k!)^{2}},}}
    0 references
    DLMF id
    DLMF:10.22.E40
    0 references
    Symbols used
    Bessel function of the second kind
    test
    Y ν ⁡ ( z ) Bessel-Y-Weber 𝜈 𝑧 {\displaystyle{\displaystyle Y_{\NVar{\nu}}\left(\NVar{z}\right)}}
    xml-id
    C10.S2.E3.m2acdec
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    Euler’s constant
    test
    γ {\displaystyle{\displaystyle\gamma}}
    xml-id
    C5.S2.E3.m2aadec
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