DLMF:5.9.E18 (Q2107)

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DLMF:5.9.E18
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    Statements

    Defining formula
    γ = - ∫ 0 ∞ e - t ⁢ ln ⁡ t ⁢ d t = ∫ 0 ∞ ( 1 1 + t - e - t ) ⁢ d t t = ∫ 0 1 ( 1 - e - t ) ⁢ d t t - ∫ 1 ∞ e - t ⁢ d t t = ∫ 0 ∞ ( e - t 1 - e - t - e - t t ) ⁢ d t . superscript subscript 0 superscript 𝑒 𝑡 𝑡 𝑡 superscript subscript 0 1 1 𝑡 superscript 𝑒 𝑡 𝑡 𝑡 superscript subscript 0 1 1 superscript 𝑒 𝑡 𝑡 𝑡 superscript subscript 1 superscript 𝑒 𝑡 𝑡 𝑡 superscript subscript 0 superscript 𝑒 𝑡 1 superscript 𝑒 𝑡 superscript 𝑒 𝑡 𝑡 𝑡 {\displaystyle{\displaystyle\gamma=-\int_{0}^{\infty}e^{-t}\ln t\mathrm{d}t=% \int_{0}^{\infty}\left(\frac{1}{1+t}-e^{-t}\right)\frac{\mathrm{d}t}{t}=\int_{% 0}^{1}(1-e^{-t})\frac{\mathrm{d}t}{t}-\int_{1}^{\infty}e^{-t}\frac{\mathrm{d}t% }{t}=\int_{0}^{\infty}\left(\frac{e^{-t}}{1-e^{-t}}-\frac{e^{-t}}{t}\right)% \mathrm{d}t.}}
    0 references
    DLMF id
    DLMF:5.9.E18
    0 references
    Symbols used
    Euler’s constant
    Defining formula
    γ {\displaystyle{\displaystyle\gamma}}
    xml-id
    C5.S2.E3.m2acdec
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    Q10770
    Defining formula
    d x 𝑥 {\displaystyle{\displaystyle\mathrm{d}\NVar{x}}}
    xml-id
    C1.S4.SS4.m1aqdec
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    base of natural logarithm
    Defining formula
    e {\displaystyle{\displaystyle\mathrm{e}}}
    xml-id
    C4.S2.E11.m2ajdec
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    Q10771
    Defining formula
    ∫ {\displaystyle{\displaystyle\int}}
    xml-id
    C1.S4.SS4.m3aqdec
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    principal branch of logarithm function
    Defining formula
    ln ⁡ z 𝑧 {\displaystyle{\displaystyle\ln\NVar{z}}}
    xml-id
    C4.S2.E2.m2addec
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