DLMF:8.4.E15 (Q2509): Difference between revisions

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Property / Symbols used
 
Property / Symbols used: Q11323 / rank
 
Normal rank
Property / Symbols used: Q11323 / qualifier
 
Defining formula:

z 𝑧 {\displaystyle{\displaystyle z}}

z
Property / Symbols used: Q11323 / qualifier
 
xml-id: C8.S1.XMD2.m1mdec

Revision as of 13:40, 2 January 2020

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DLMF:8.4.E15
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    Statements

    Defining formula
    Γ ⁡ ( - n , z ) = ( - 1 ) n n ! ⁢ ( E 1 ⁡ ( z ) - e - z ⁢ ∑ k = 0 n - 1 ( - 1 ) k ⁢ k ! z k + 1 ) = ( - 1 ) n n ! ⁢ ( ψ ⁡ ( n + 1 ) - ln ⁡ z ) - z - n ⁢ ∑ k = 0 k ≠ n ∞ ( - z ) k k ! ⁢ ( k - n ) . incomplete-Gamma 𝑛 𝑧 superscript 1 𝑛 𝑛 exponential-integral 𝑧 superscript 𝑒 𝑧 superscript subscript 𝑘 0 𝑛 1 superscript 1 𝑘 𝑘 superscript 𝑧 𝑘 1 superscript 1 𝑛 𝑛 digamma 𝑛 1 𝑧 superscript 𝑧 𝑛 superscript subscript 𝑘 0 𝑘 𝑛 superscript 𝑧 𝑘 𝑘 𝑘 𝑛 {\displaystyle{\displaystyle\Gamma\left(-n,z\right)=\frac{(-1)^{n}}{n!}\left(E% _{1}\left(z\right)-e^{-z}\sum_{k=0}^{n-1}\frac{(-1)^{k}k!}{z^{k+1}}\right)=% \frac{(-1)^{n}}{n!}\left(\psi\left(n+1\right)-\ln z\right)-z^{-n}\sum_{% \begin{subarray}{c}k=0\\ k\neq n\end{subarray}}^{\infty}\frac{(-z)^{k}}{k!(k-n)}.}}
    0 references
    DLMF id
    DLMF:8.4.E15
    0 references
    Symbols used
    psi (or digamma) function
    Defining formula
    ψ ⁡ ( z ) digamma 𝑧 {\displaystyle{\displaystyle\psi\left(\NVar{z}\right)}}
    xml-id
    C5.S2.E2.m2adec
    0 references
    base of natural logarithm
    Defining formula
    e {\displaystyle{\displaystyle\mathrm{e}}}
    xml-id
    C4.S2.E11.m2ajdec
    0 references
    exponential integral
    Defining formula
    E 1 ⁡ ( z ) exponential-integral 𝑧 {\displaystyle{\displaystyle E_{1}\left(\NVar{z}\right)}}
    xml-id
    C6.S2.E1.m2aadec
    0 references
    Q10755
    Defining formula
    ! {\displaystyle{\displaystyle!}}
    xml-id
    introduction.Sx4.p1.t1.r15.m5acdec
    0 references
    incomplete gamma function
    Defining formula
    Γ ⁡ ( a , z ) incomplete-Gamma 𝑎 𝑧 {\displaystyle{\displaystyle\Gamma\left(\NVar{a},\NVar{z}\right)}}
    xml-id
    C8.S2.E2.m2aedec
    0 references
    principal branch of logarithm function
    Defining formula
    ln ⁡ z 𝑧 {\displaystyle{\displaystyle\ln\NVar{z}}}
    xml-id
    C4.S2.E2.m2adec
    0 references
    Q11323
    Defining formula
    z 𝑧 {\displaystyle{\displaystyle z}}
    xml-id
    C8.S1.XMD2.m1mdec
    0 references
     
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