DLMF:5.9.E5 (Q2094)

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DLMF:5.9.E5
No description defined

    Statements

    Defining formula
    Γ ⁡ ( z ) = ∫ 0 ∞ t z - 1 ⁢ ( e - t - ∑ k = 0 n ( - 1 ) k ⁢ t k k ! ) ⁢ d t , Euler-Gamma 𝑧 superscript subscript 0 superscript 𝑡 𝑧 1 superscript 𝑒 𝑡 superscript subscript 𝑘 0 𝑛 superscript 1 𝑘 superscript 𝑡 𝑘 𝑘 𝑡 {\displaystyle{\displaystyle\Gamma\left(z\right)=\int_{0}^{\infty}t^{z-1}\left% (e^{-t}-\sum_{k=0}^{n}\frac{(-1)^{k}t^{k}}{k!}\right)\mathrm{d}t,}}
    0 references
    DLMF id
    DLMF:5.9.E5
    0 references
    constraint
    - n - 1 < ℜ ⁡ z < - n 𝑛 1 𝑧 𝑛 {\displaystyle{\displaystyle-n-1<\Re z<-n}}
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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.m2addec
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    Q10770
    Defining formula
    d x 𝑥 {\displaystyle{\displaystyle\mathrm{d}\NVar{x}}}
    xml-id
    C1.S4.SS4.m1addec
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    base of natural logarithm
    Defining formula
    e {\displaystyle{\displaystyle\mathrm{e}}}
    xml-id
    C4.S2.E11.m2acdec
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    Q10755
    Defining formula
    ! {\displaystyle{\displaystyle!}}
    xml-id
    introduction.Sx4.p1.t1.r15.m5aadec
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    Q10771
    Defining formula
    ∫ {\displaystyle{\displaystyle\int}}
    xml-id
    C1.S4.SS4.m3addec
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    Q10811
    Defining formula
    ℜ ⁡ absent {\displaystyle{\displaystyle\Re}}
    xml-id
    C1.S9.E2.m1aadec
    0 references
    Q11274
    Defining formula
    n 𝑛 {\displaystyle{\displaystyle n}}
    xml-id
    C5.S1.XMD3.m1dec
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    Q11279
    Defining formula
    k 𝑘 {\displaystyle{\displaystyle k}}
    xml-id
    C5.S1.XMD4.m1adec
    0 references
    Q11273
    Defining formula
    z 𝑧 {\displaystyle{\displaystyle z}}
    xml-id
    C5.S1.XMD7.m1ddec
    0 references
     
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