DLMF:5.9.E4 (Q2093)

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DLMF:5.9.E4
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    Statements

    Defining formula
    Γ ⁡ ( z ) = ∫ 1 ∞ t z - 1 ⁢ e - t ⁢ d t + ∑ k = 0 ∞ ( - 1 ) k ( z + k ) ⁢ k ! , Euler-Gamma 𝑧 superscript subscript 1 superscript 𝑡 𝑧 1 superscript 𝑒 𝑡 𝑡 superscript subscript 𝑘 0 superscript 1 𝑘 𝑧 𝑘 𝑘 {\displaystyle{\displaystyle\Gamma\left(z\right)=\int_{1}^{\infty}t^{z-1}e^{-t% }\mathrm{d}t+\sum_{k=0}^{\infty}\frac{(-1)^{k}}{(z+k)k!},}}
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    DLMF id
    DLMF:5.9.E4
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    constraint
    z ≠ 0 , - 1 , - 2 , … 𝑧 0 1 2 … {\displaystyle{\displaystyle z\neq 0,-1,-2,\dots}}
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    Symbols used
    gamma function
    Defining formula
    Γ ⁡ ( z ) Euler-Gamma 𝑧 {\displaystyle{\displaystyle\Gamma\left(\NVar{z}\right)}}
    xml-id
    C5.S2.E1.m2acdec
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    Q10770
    Defining formula
    d x 𝑥 {\displaystyle{\displaystyle\mathrm{d}\NVar{x}}}
    xml-id
    C1.S4.SS4.m1acdec
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    base of natural logarithm
    Defining formula
    e {\displaystyle{\displaystyle\mathrm{e}}}
    xml-id
    C4.S2.E11.m2abdec
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    Q10755
    Defining formula
    ! {\displaystyle{\displaystyle!}}
    xml-id
    introduction.Sx4.p1.t1.r15.m5adec
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    Q10771
    Defining formula
    ∫ {\displaystyle{\displaystyle\int}}
    xml-id
    C1.S4.SS4.m3acdec
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