DLMF:8.18.E3 (Q2669): Difference between revisions

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Property / Symbols used
 
Property / Symbols used: Q10931 / rank
 
Normal rank
Property / Symbols used: Q10931 / qualifier
 
Defining formula:

∼ asymptotic-expansion {\displaystyle{\displaystyle\sim}}

\asympexp
Property / Symbols used: Q10931 / qualifier
 
xml-id: C2.S1.SS3.p1.m11adec
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
Property / Symbols used
 
Property / Symbols used: Q11329 / rank
 
Normal rank
Property / Symbols used: Q11329 / qualifier
 
Defining formula:

n 𝑛 {\displaystyle{\displaystyle n}}

n
Property / Symbols used: Q11329 / qualifier
 
xml-id: C8.S1.XMD6.m1adec
Property / Symbols used
 
Property / Symbols used: Q11360 / rank
 
Normal rank
Property / Symbols used: Q11360 / qualifier
 
Defining formula:

a 𝑎 {\displaystyle{\displaystyle a}}

a
Property / Symbols used: Q11360 / qualifier
 
xml-id: C8.S17.XMD1.m1adec
Property / Symbols used
 
Property / Symbols used: Q11361 / rank
 
Normal rank
Property / Symbols used: Q11361 / qualifier
 
Defining formula:

b 𝑏 {\displaystyle{\displaystyle b}}

b
Property / Symbols used: Q11361 / qualifier
 
xml-id: C8.S17.XMD2.m1adec
Property / Symbols used
 
Property / Symbols used: Q11362 / rank
 
Normal rank
Property / Symbols used: Q11362 / qualifier
 
Defining formula:

x 𝑥 {\displaystyle{\displaystyle x}}

x
Property / Symbols used: Q11362 / qualifier
 
xml-id: C8.S17.XMD3.m1bdec
Property / Symbols used
 
Property / Symbols used: Q11371 / rank
 
Normal rank
Property / Symbols used: Q11371 / qualifier
 
Defining formula:

F k subscript 𝐹 𝑘 {\displaystyle{\displaystyle F_{k}}}

F_{k}
Property / Symbols used: Q11371 / qualifier
 
xml-id: C8.S18.XMD2.m1dec
Property / Symbols used
 
Property / Symbols used: Q11372 / rank
 
Normal rank
Property / Symbols used: Q11372 / qualifier
 
Defining formula:

d k subscript 𝑑 𝑘 {\displaystyle{\displaystyle d_{k}}}

d_{k}
Property / Symbols used: Q11372 / qualifier
 
xml-id: C8.S18.XMD3.m1dec

Latest revision as of 14:06, 2 January 2020

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DLMF:8.18.E3
No description defined

    Statements

    Defining formula
    I x ⁡ ( a , b ) = Γ ⁡ ( a + b ) Γ ⁡ ( a ) ⁢ ( ∑ k = 0 n - 1 d k ⁢ F k + O ⁡ ( a - n ) ⁢ F 0 ) , IncI 𝑥 𝑎 𝑏 Euler-Gamma 𝑎 𝑏 Euler-Gamma 𝑎 superscript subscript 𝑘 0 𝑛 1 subscript 𝑑 𝑘 subscript 𝐹 𝑘 Big-O superscript 𝑎 𝑛 subscript 𝐹 0 {\displaystyle{\displaystyle I_{x}\left(a,b\right)=\frac{\Gamma\left(a+b\right% )}{\Gamma\left(a\right)}\left(\sum_{k=0}^{n-1}d_{k}F_{k}+O\left(a^{-n}\right)F% _{0}\right),}}
    0 references
    DLMF id
    DLMF:8.18.E3
    0 references
    Symbols used
    order not exceeding
    Defining formula
    O ⁡ ( x ) Big-O 𝑥 {\displaystyle{\displaystyle O\left(\NVar{x}\right)}}
    xml-id
    C2.S1.E3.m2aadec
    0 references
    gamma function
    Defining formula
    Γ ⁡ ( z ) Euler-Gamma 𝑧 {\displaystyle{\displaystyle\Gamma\left(\NVar{z}\right)}}
    xml-id
    C5.S2.E1.m2aadec
    0 references
    incomplete beta function
    Defining formula
    I x ⁡ ( a , b ) IncI 𝑥 𝑎 𝑏 {\displaystyle{\displaystyle I_{\NVar{x}}\left(\NVar{a},\NVar{b}\right)}}
    xml-id
    C8.S17.E2.m2aadec
    0 references
    Q10931
    Defining formula
    ∼ asymptotic-expansion {\displaystyle{\displaystyle\sim}}
    xml-id
    C2.S1.SS3.p1.m11adec
    0 references
    Q11330
    Defining formula
    k 𝑘 {\displaystyle{\displaystyle k}}
    xml-id
    C8.S1.XMD5.m1adec
    0 references
    Q11329
    Defining formula
    n 𝑛 {\displaystyle{\displaystyle n}}
    xml-id
    C8.S1.XMD6.m1adec
    0 references
    Q11360
    Defining formula
    a 𝑎 {\displaystyle{\displaystyle a}}
    xml-id
    C8.S17.XMD1.m1adec
    0 references
    Q11361
    Defining formula
    b 𝑏 {\displaystyle{\displaystyle b}}
    xml-id
    C8.S17.XMD2.m1adec
    0 references
    Q11362
    Defining formula
    x 𝑥 {\displaystyle{\displaystyle x}}
    xml-id
    C8.S17.XMD3.m1bdec
    0 references
    Q11371
    Defining formula
    F k subscript 𝐹 𝑘 {\displaystyle{\displaystyle F_{k}}}
    xml-id
    C8.S18.XMD2.m1dec
    0 references
    Q11372
    Defining formula
    d k subscript 𝑑 𝑘 {\displaystyle{\displaystyle d_{k}}}
    xml-id
    C8.S18.XMD3.m1dec
    0 references
     
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