DLMF:8.19.E25 (Q2711)

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DLMF:8.19.E25
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    Statements

    test
    ∫ 0 ∞ e - a ⁢ t ⁢ t b - 1 ⁢ E p ⁡ ( t ) ⁢ d t = Γ ⁡ ( b ) ⁢ ( 1 + a ) - b p + b - 1 ⁢ F ⁡ ( 1 , b ; p + b ; a / ( 1 + a ) ) , superscript subscript 0 superscript 𝑒 𝑎 𝑡 superscript 𝑡 𝑏 1 exponential-integral-En 𝑝 𝑡 𝑡 Euler-Gamma 𝑏 superscript 1 𝑎 𝑏 𝑝 𝑏 1 Gauss-hypergeometric-F 1 𝑏 𝑝 𝑏 𝑎 1 𝑎 {\displaystyle{\displaystyle\int_{0}^{\infty}e^{-at}t^{b-1}E_{p}\left(t\right)% \mathrm{d}t=\frac{\Gamma\left(b\right)(1+a)^{-b}}{p+b-1}\*F\left(1,b;p+b;a/(1+% a)\right),}}
    0 references
    DLMF id
    DLMF:8.19.E25
    0 references
    constraint
    ℜ ⁡ a > - 1 𝑎 1 {\displaystyle{\displaystyle\Re a>-1}}
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    ℜ ⁡ ( p + b ) > 1 𝑝 𝑏 1 {\displaystyle{\displaystyle\Re(p+b)>1}}
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    Symbols used
    gamma function
    test
    Γ ⁡ ( z ) Euler-Gamma 𝑧 {\displaystyle{\displaystyle\Gamma\left(\NVar{z}\right)}}
    xml-id
    C5.S2.E1.m2aedec
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    $$={{}_{2}F_{1}}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$$ Gauss’ hypergeometric function
    test
    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.m2adec
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    Q10770
    test
    d x 𝑥 {\displaystyle{\displaystyle\mathrm{d}\NVar{x}}}
    xml-id
    C1.S4.SS4.m1afdec
    0 references
    base of natural logarithm
    test
    e {\displaystyle{\displaystyle\mathrm{e}}}
    xml-id
    C4.S2.E11.m2aodec
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    generalized exponential integral
    test
    E p ⁡ ( z ) exponential-integral-En 𝑝 𝑧 {\displaystyle{\displaystyle E_{\NVar{p}}\left(\NVar{z}\right)}}
    xml-id
    C8.S19.E1.m2aacdec
    0 references
    Q10771
    test
    ∫ {\displaystyle{\displaystyle\int}}
    xml-id
    C1.S4.SS4.m3afdec
    0 references
    Q10811
    test
    ℜ ⁡ absent {\displaystyle{\displaystyle\Re}}
    xml-id
    C1.S9.E2.m1addec
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    Q11324
    test
    a 𝑎 {\displaystyle{\displaystyle a}}
    xml-id
    C8.S1.XMD3.m1adec
    0 references
    Q11377
    test
    p 𝑝 {\displaystyle{\displaystyle p}}
    xml-id
    C8.S1.XMD4.m1pdec
    0 references
     
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