DLMF:8.14.E6 (Q2639): Difference between revisions

From DRMF
Jump to navigation Jump to search
← Older editNewer edit →
‎Changed an Item: Add constraint
‎Changed an Item: Add constraint
Property / Symbols used
 
Property / Symbols used: $$={{}_{2}F_{1}}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$$ Gauss’ hypergeometric function / rank
 
Normal rank
Property / Symbols used: $$={{}_{2}F_{1}}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$$ Gauss’ hypergeometric function / qualifier
 
Defining formula:

F ⁡ ( a , b ; c ; z ) Gauss-hypergeometric-F 𝑎 𝑏 𝑐 𝑧 {\displaystyle{\displaystyle F\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)}}

\hyperF@{\NVar{a}}{\NVar{b}}{\NVar{c}}{\NVar{z}}
Property / Symbols used: $$={{}_{2}F_{1}}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$$ Gauss’ hypergeometric function / qualifier
 
xml-id: C15.S2.E1.m2aadec

Revision as of 14:01, 2 January 2020

No description defined
Language Label Description Also known as
English
DLMF:8.14.E6
No description defined

    Statements

    Defining formula
    ∫ 0 ∞ x a - 1 ⁢ e - s ⁢ x ⁢ Γ ⁡ ( b , x ) ⁢ d x = Γ ⁡ ( a + b ) a ⁢ ( 1 + s ) a + b ⁢ F ⁡ ( 1 , a + b ; 1 + a ; s / ( 1 + s ) ) , superscript subscript 0 superscript 𝑥 𝑎 1 superscript 𝑒 𝑠 𝑥 incomplete-Gamma 𝑏 𝑥 𝑥 Euler-Gamma 𝑎 𝑏 𝑎 superscript 1 𝑠 𝑎 𝑏 Gauss-hypergeometric-F 1 𝑎 𝑏 1 𝑎 𝑠 1 𝑠 {\displaystyle{\displaystyle\int_{0}^{\infty}x^{a-1}e^{-sx}\Gamma\left(b,x% \right)\mathrm{d}x=\frac{\Gamma\left(a+b\right)}{a(1+s)^{a+b}}\*F\left(1,a+b;1% +a;s/(1+s)\right),}}
    0 references
    DLMF id
    DLMF:8.14.E6
    0 references
    constraint
    ℜ ⁡ s > - 1 𝑠 1 {\displaystyle{\displaystyle\Re s>-1}}
    0 references
    ℜ ⁡ ( a + b ) > 0 𝑎 𝑏 0 {\displaystyle{\displaystyle\Re(a+b)>0}}
    0 references
    ℜ ⁡ a > 0 𝑎 0 {\displaystyle{\displaystyle\Re a>0}}
    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.m2aedec
    0 references
    $$={{}_{2}F_{1}}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$$ Gauss’ hypergeometric function
    Defining formula
    F ⁡ ( a , b ; c ; z ) Gauss-hypergeometric-F 𝑎 𝑏 𝑐 𝑧 {\displaystyle{\displaystyle F\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)}}
    xml-id
    C15.S2.E1.m2aadec
    0 references
     
        Retrieved from "https://drmf-beta.wmflabs.org/index.php?title=Item:Q2639&oldid=42588"