DLMF:10.22.E39 (Q3413)

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DLMF:10.22.E39
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    Statements

    Defining formula
    ∫ x ∞ J 0 ⁡ ( t ) t ⁢ d t + γ + ln ⁡ ( 1 2 ⁢ x ) = ∫ 0 x 1 - J 0 ⁡ ( t ) t ⁢ d t = ∑ k = 1 ∞ ( - 1 ) k - 1 ⁢ ( 1 2 ⁢ x ) 2 ⁢ k 2 ⁢ k ⁢ ( k ! ) 2 , superscript subscript 𝑥 Bessel-J 0 𝑡 𝑡 𝑡 1 2 𝑥 superscript subscript 0 𝑥 1 Bessel-J 0 𝑡 𝑡 𝑡 superscript subscript 𝑘 1 superscript 1 𝑘 1 superscript 1 2 𝑥 2 𝑘 2 𝑘 superscript 𝑘 2 {\displaystyle{\displaystyle\int_{x}^{\infty}\frac{J_{0}\left(t\right)}{t}% \mathrm{d}t+\gamma+\ln\left(\tfrac{1}{2}x\right)=\int_{0}^{x}\frac{1-J_{0}% \left(t\right)}{t}\mathrm{d}t=\sum_{k=1}^{\infty}(-1)^{k-1}\frac{(\frac{1}{2}x% )^{2k}}{2k(k!)^{2}},}}
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    DLMF id
    DLMF:10.22.E39
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