Deprecated: $wgMWOAuthSharedUserIDs=false is deprecated, set $wgMWOAuthSharedUserIDs=true, $wgMWOAuthSharedUserSource='local' instead [Called from MediaWiki\HookContainer\HookContainer::run in /var/www/html/w/includes/HookContainer/HookContainer.php at line 135] in /var/www/html/w/includes/Debug/MWDebug.php on line 372
DLMF:8.18.E1 - MaRDI portal

DLMF:8.18.E1 (Q3365)

From MaRDI portal





No description defined
Language Label Description Also known as
English
DLMF:8.18.E1
No description defined

    Statements

    DLMF defining formula
    I x ⁡ ( a , b ) = Γ ⁡ ( a + b ) ⁢ x a ⁢ ( 1 - x ) b - 1 ⁢ ( ∑ k = 0 n - 1 1 Γ ⁡ ( a + k + 1 ) ⁢ Γ ⁡ ( b - k ) ⁢ ( x 1 - x ) k + O ⁡ ( 1 Γ ⁡ ( a + n + 1 ) ) ) , IncI 𝑥 𝑎 𝑏 Euler-Gamma 𝑎 𝑏 superscript 𝑥 𝑎 superscript 1 𝑥 𝑏 1 superscript subscript 𝑘 0 𝑛 1 1 Euler-Gamma 𝑎 𝑘 1 Euler-Gamma 𝑏 𝑘 superscript 𝑥 1 𝑥 𝑘 Big-O 1 Euler-Gamma 𝑎 𝑛 1 {\displaystyle{\displaystyle I_{x}\left(a,b\right)={\Gamma\left(a+b\right)x^{a% }(1-x)^{b-1}}\*\left(\sum_{k=0}^{n-1}\frac{1}{\Gamma\left(a+k+1\right)\Gamma% \left(b-k\right)}\left(\frac{x}{1-x}\right)^{k}+O\left(\frac{1}{\Gamma\left(a+% n+1\right)}\right)\right),}}
    0 references
    Symbols used
    order not exceeding
    DLMF defining formula
    O ⁡ ( x ) Big-O 𝑥 {\displaystyle{\displaystyle O\left(\NVar{x}\right)}}
    xml-id
    C2.S1.E3.m2adec
    0 references
    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
    incomplete beta function
    DLMF 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.m2adec
    0 references
    nonnegative integer
    DLMF defining formula
    k 𝑘 {\displaystyle{\displaystyle k}}
    xml-id
    C8.S1.XMD5.m1dec
    0 references
    nonnegative integer
    DLMF defining formula
    n 𝑛 {\displaystyle{\displaystyle n}}
    xml-id
    C8.S1.XMD6.m1dec
    0 references
    parameter
    DLMF defining formula
    a 𝑎 {\displaystyle{\displaystyle a}}
    xml-id
    C8.S17.XMD1.m1dec
    0 references
    parameter
    DLMF defining formula
    b 𝑏 {\displaystyle{\displaystyle b}}
    xml-id
    C8.S17.XMD2.m1dec
    0 references
    variable
    DLMF defining formula
    x 𝑥 {\displaystyle{\displaystyle x}}
    xml-id
    C8.S17.XMD3.m1dec
    0 references

    Identifiers

     
    Retrieved from "https://portal.mardi4nfdi.de/w/index.php?title=Item:Q3365&oldid=24165497"
    Powered by MediaWiki