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DLMF:8.7.E1 - MaRDI portal

DLMF:8.7.E1 (Q3225)

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DLMF:8.7.E1
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    Statements

    DLMF defining formula
    γ * ⁡ ( a , z ) = e - z ⁢ ∑ k = 0 ∞ z k Γ ⁡ ( a + k + 1 ) = 1 Γ ⁡ ( a ) ⁢ ∑ k = 0 ∞ ( - z ) k k ! ⁢ ( a + k ) . incomplete-gamma-star 𝑎 𝑧 superscript 𝑒 𝑧 superscript subscript 𝑘 0 superscript 𝑧 𝑘 Euler-Gamma 𝑎 𝑘 1 1 Euler-Gamma 𝑎 superscript subscript 𝑘 0 superscript 𝑧 𝑘 𝑘 𝑎 𝑘 {\displaystyle{\displaystyle\gamma^{*}\left(a,z\right)=e^{-z}\sum_{k=0}^{% \infty}\frac{z^{k}}{\Gamma\left(a+k+1\right)}=\frac{1}{\Gamma\left(a\right)}% \sum_{k=0}^{\infty}\frac{(-z)^{k}}{k!(a+k)}.}}
    0 references
    Symbols used
    gamma function
    DLMF defining formula
    Γ ⁡ ( z ) Euler-Gamma 𝑧 {\displaystyle{\displaystyle\Gamma\left(\NVar{z}\right)}}
    xml-id
    C5.S2.E1.m2adec
    0 references
    base of natural logarithm
    DLMF defining formula
    e {\displaystyle{\displaystyle\mathrm{e}}}
    xml-id
    C4.S2.E11.m2adec
    0 references
    Q10729
    DLMF defining formula
    ! {\displaystyle{\displaystyle!}}
    xml-id
    introduction.Sx4.p1.t1.r15.m5adec
    0 references
    incomplete gamma function
    DLMF defining formula
    γ * ⁡ ( a , z ) incomplete-gamma-star 𝑎 𝑧 {\displaystyle{\displaystyle\gamma^{*}\left(\NVar{a},\NVar{z}\right)}}
    xml-id
    C8.S2.E6.m2adec
    0 references
    complex variable
    DLMF defining formula
    z 𝑧 {\displaystyle{\displaystyle z}}
    xml-id
    C8.S1.XMD2.m1dec
    0 references
    parameter
    DLMF defining formula
    a 𝑎 {\displaystyle{\displaystyle a}}
    xml-id
    C8.S1.XMD3.m1dec
    0 references
    nonnegative integer
    DLMF defining formula
    k 𝑘 {\displaystyle{\displaystyle k}}
    xml-id
    C8.S1.XMD5.m1dec
    0 references

    Identifiers

     
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