DLMF:8.7.E3 (Q2529): 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.m1adec
Property / Symbols used
 
Property / Symbols used: Q11324 / rank
 
Normal rank
Property / Symbols used: Q11324 / qualifier
 
Defining formula:

a 𝑎 {\displaystyle{\displaystyle a}}

a
Property / Symbols used: Q11324 / qualifier
 
xml-id: C8.S1.XMD3.m1bdec
Property / Symbols used
 
Property / Symbols used: Q11330 / rank
 
Normal rank
Property / Symbols used: Q11330 / qualifier
 
Defining formula:

k 𝑘 {\displaystyle{\displaystyle k}}

k
Property / Symbols used: Q11330 / qualifier
 
xml-id: C8.S1.XMD5.m1adec

Latest revision as of 13:45, 2 January 2020

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DLMF:8.7.E3
No description defined

    Statements

    Defining formula
    Γ ⁡ ( a , z ) = Γ ⁡ ( a ) - ∑ k = 0 ∞ ( - 1 ) k ⁢ z a + k k ! ⁢ ( a + k ) = Γ ⁡ ( a ) ⁢ ( 1 - z a ⁢ e - z ⁢ ∑ k = 0 ∞ z k Γ ⁡ ( a + k + 1 ) ) , incomplete-Gamma 𝑎 𝑧 Euler-Gamma 𝑎 superscript subscript 𝑘 0 superscript 1 𝑘 superscript 𝑧 𝑎 𝑘 𝑘 𝑎 𝑘 Euler-Gamma 𝑎 1 superscript 𝑧 𝑎 superscript 𝑒 𝑧 superscript subscript 𝑘 0 superscript 𝑧 𝑘 Euler-Gamma 𝑎 𝑘 1 {\displaystyle{\displaystyle\Gamma\left(a,z\right)=\Gamma\left(a\right)-\sum_{% k=0}^{\infty}\frac{(-1)^{k}z^{a+k}}{k!(a+k)}=\Gamma\left(a\right)\left(1-z^{a}% e^{-z}\sum_{k=0}^{\infty}\frac{z^{k}}{\Gamma\left(a+k+1\right)}\right),}}
    0 references
    DLMF id
    DLMF:8.7.E3
    0 references
    constraint
    a ≠ 0 , - 1 , - 2 , … 𝑎 0 1 2 … {\displaystyle{\displaystyle a\neq 0,-1,-2,\dots}}
    0 references
    Symbols used
    gamma function
    Defining formula
    Γ ⁡ ( z ) Euler-Gamma 𝑧 {\displaystyle{\displaystyle\Gamma\left(\NVar{z}\right)}}
    xml-id
    C5.S2.E1.m2aadec
    0 references
    base of natural logarithm
    Defining formula
    e {\displaystyle{\displaystyle\mathrm{e}}}
    xml-id
    C4.S2.E11.m2abdec
    0 references
    Q10755
    Defining formula
    ! {\displaystyle{\displaystyle!}}
    xml-id
    introduction.Sx4.p1.t1.r15.m5aadec
    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.m2aadec
    0 references
    Q11323
    Defining formula
    z 𝑧 {\displaystyle{\displaystyle z}}
    xml-id
    C8.S1.XMD2.m1adec
    0 references
    Q11324
    Defining formula
    a 𝑎 {\displaystyle{\displaystyle a}}
    xml-id
    C8.S1.XMD3.m1bdec
    0 references
    Q11330
    Defining formula
    k 𝑘 {\displaystyle{\displaystyle k}}
    xml-id
    C8.S1.XMD5.m1adec
    0 references
     
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