DLMF:13.10.E7 (Q4466): Difference between revisions

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Property / Symbols used
 
Property / Symbols used: Q10771 / rank
 
Normal rank
Property / Symbols used: Q10771 / qualifier
 
Defining formula:

∫ {\displaystyle{\displaystyle\int}}

\int
Property / Symbols used: Q10771 / qualifier
 
xml-id: C1.S4.SS4.m3afdec
Property / Symbols used
 
Property / Symbols used: Q11557 / rank
 
Normal rank
Property / Symbols used: Q11557 / qualifier
 
Defining formula:

z 𝑧 {\displaystyle{\displaystyle z}}

z
Property / Symbols used: Q11557 / qualifier
 
xml-id: C13.S1.XMD6.m1edec
Property / Symbols used
 
Property / Symbols used: Q10811 / rank
 
Normal rank
Property / Symbols used: Q10811 / qualifier
 
Defining formula:

ℜ ⁡ absent {\displaystyle{\displaystyle\Re}}

\realpart@@
Property / Symbols used: Q10811 / qualifier
 
xml-id: C1.S9.E2.m1addec

Latest revision as of 16:01, 2 January 2020

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DLMF:13.10.E7
No description defined

    Statements

    Defining formula
    ∫ 0 ∞ e - z ⁢ t ⁢ t b - 1 ⁢ U ⁡ ( a , c , t ) ⁢ d t = Γ ⁡ ( b ) ⁢ Γ ⁡ ( b - c + 1 ) ⁢ z - b ⁢ 𝐅 1 2 ⁡ ( a , b ; a + b - c + 1 ; 1 - 1 z ) , superscript subscript 0 superscript 𝑒 𝑧 𝑡 superscript 𝑡 𝑏 1 Kummer-confluent-hypergeometric-U 𝑎 𝑐 𝑡 𝑡 Euler-Gamma 𝑏 Euler-Gamma 𝑏 𝑐 1 superscript 𝑧 𝑏 hypergeometric-bold-pFq 2 1 𝑎 𝑏 𝑎 𝑏 𝑐 1 1 1 𝑧 {\displaystyle{\displaystyle\int_{0}^{\infty}e^{-zt}t^{b-1}U\left(a,c,t\right)% \mathrm{d}t=\Gamma\left(b\right)\Gamma\left(b-c+1\right)\*z^{-b}{{}_{2}{% \mathbf{F}}_{1}}\left(a,b;a+b-c+1;1-\frac{1}{z}\right),}}
    0 references
    DLMF id
    DLMF:13.10.E7
    0 references
    constraint
    ℜ ⁡ b > max ⁡ ( ℜ ⁡ c - 1 , 0 ) 𝑏 𝑐 1 0 {\displaystyle{\displaystyle\Re b>\max\left(\Re c-1,0\right)}}
    0 references
    ℜ ⁡ z > 0 𝑧 0 {\displaystyle{\displaystyle\Re z>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.m2acdec
    0 references
    Kummer confluent hypergeometric function
    Defining formula
    U ⁡ ( a , b , z ) Kummer-confluent-hypergeometric-U 𝑎 𝑏 𝑧 {\displaystyle{\displaystyle U\left(\NVar{a},\NVar{b},\NVar{z}\right)}}
    xml-id
    C13.S2.E6.m2abdec
    0 references
    Q10770
    Defining formula
    d x 𝑥 {\displaystyle{\displaystyle\mathrm{d}\NVar{x}}}
    xml-id
    C1.S4.SS4.m1afdec
    0 references
    base of natural logarithm
    Defining formula
    e {\displaystyle{\displaystyle\mathrm{e}}}
    xml-id
    C4.S2.E11.m2addec
    0 references
    scaled (or Olver’s) generalized hypergeometric function
    Defining formula
    𝐅 q p ⁡ ( 𝐚 ; 𝐛 ; ) hypergeometric-bold-pFq 𝑝 𝑞 𝐚 𝐛 {\displaystyle{\displaystyle{{}_{\NVar{p}}{\mathbf{F}}_{\NVar{q}}}\left(\NVar{% \mathbf{a}};\NVar{\mathbf{b}};\right)\)\@add@PDF@RDFa@triples\end{document}}}
    xml-id
    C16.S2.E5.m2aadec
    0 references
    Q10771
    Defining formula
    ∫ {\displaystyle{\displaystyle\int}}
    xml-id
    C1.S4.SS4.m3afdec
    0 references
    Q11557
    Defining formula
    z 𝑧 {\displaystyle{\displaystyle z}}
    xml-id
    C13.S1.XMD6.m1edec
    0 references
    Q10811
    Defining formula
    ℜ ⁡ absent {\displaystyle{\displaystyle\Re}}
    xml-id
    C1.S9.E2.m1addec
    0 references
     
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