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DLMF:10.22.E39 - MaRDI portal

DLMF:10.22.E39 (Q4110)

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DLMF:10.22.E39
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    Statements

    DLMF 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}},}}
    0 references
    Symbols used
    Bessel function of the first kind
    DLMF defining formula
    J ν ⁡ ( z ) Bessel-J 𝜈 𝑧 {\displaystyle{\displaystyle J_{\NVar{\nu}}\left(\NVar{z}\right)}}
    xml-id
    C10.S2.E2.m2aaedec
    0 references
    Euler’s constant
    DLMF defining formula
    γ {\displaystyle{\displaystyle\gamma}}
    xml-id
    C5.S2.E3.m2adec
    0 references
    differential of x
    DLMF defining formula
    d x 𝑥 {\displaystyle{\displaystyle\mathrm{d}\NVar{x}}}
    xml-id
    C1.S4.SS4.m1aaldec
    0 references
    Q10729
    DLMF defining formula
    ! {\displaystyle{\displaystyle!}}
    xml-id
    introduction.Sx4.p1.t1.r15.m5abdec
    0 references
    integral
    DLMF defining formula
    ∫ {\displaystyle{\displaystyle\int}}
    xml-id
    C1.S4.SS4.m3aaldec
    0 references
    principal branch of logarithm function
    DLMF defining formula
    ln ⁡ z 𝑧 {\displaystyle{\displaystyle\ln\NVar{z}}}
    xml-id
    C4.S2.E2.m2adec
    0 references
    nonnegative integer
    DLMF defining formula
    k 𝑘 {\displaystyle{\displaystyle k}}
    xml-id
    C10.S1.XMD3.m1idec
    0 references
    real variable
    DLMF defining formula
    x 𝑥 {\displaystyle{\displaystyle x}}
    xml-id
    C10.S1.XMD4.m1odec
    0 references

    Identifiers

     
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