DLMF:9.10.E15 (Q2898)

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DLMF:9.10.E15
No description defined

    Statements

    Defining formula
    ∫ 0 ∞ e - p ⁢ t ⁢ Ai ⁡ ( - t ) ⁢ d t = 1 3 ⁢ e p 3 / 3 ⁢ ( Γ ⁡ ( 1 3 , 1 3 ⁢ p 3 ) Γ ⁡ ( 1 3 ) + Γ ⁡ ( 2 3 , 1 3 ⁢ p 3 ) Γ ⁡ ( 2 3 ) ) , superscript subscript 0 superscript 𝑒 𝑝 𝑡 Airy-Ai 𝑡 𝑡 1 3 superscript 𝑒 superscript 𝑝 3 3 incomplete-Gamma 1 3 1 3 superscript 𝑝 3 Euler-Gamma 1 3 incomplete-Gamma 2 3 1 3 superscript 𝑝 3 Euler-Gamma 2 3 {\displaystyle{\displaystyle\int_{0}^{\infty}e^{-pt}\mathrm{Ai}\left(-t\right)% \mathrm{d}t={\frac{1}{3}e^{p^{3}/3}\left(\frac{\Gamma\left(\tfrac{1}{3},\tfrac% {1}{3}p^{3}\right)}{\Gamma\left(\tfrac{1}{3}\right)}+\frac{\Gamma\left(\tfrac{% 2}{3},\tfrac{1}{3}p^{3}\right)}{\Gamma\left(\tfrac{2}{3}\right)}\right)},}}
    0 references
    DLMF id
    DLMF:9.10.E15
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    constraint
    ℜ ⁡ p > 0 𝑝 0 {\displaystyle{\displaystyle\Re p>0}}
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    Symbols used
    Q10919
    Defining formula
    Ai ⁡ ( z ) Airy-Ai 𝑧 {\displaystyle{\displaystyle\mathrm{Ai}\left(\NVar{z}\right)}}
    xml-id
    C9.S2.SS1.m1agdec
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    gamma function
    Defining formula
    Γ ⁡ ( z ) Euler-Gamma 𝑧 {\displaystyle{\displaystyle\Gamma\left(\NVar{z}\right)}}
    xml-id
    C5.S2.E1.m2aadec
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    Q10770
    Defining formula
    d x 𝑥 {\displaystyle{\displaystyle\mathrm{d}\NVar{x}}}
    xml-id
    C1.S4.SS4.m1andec
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    base of natural logarithm
    Defining formula
    e {\displaystyle{\displaystyle\mathrm{e}}}
    xml-id
    C4.S2.E11.m2abdec
    0 references
    incomplete gamma function
    Defining formula
    Γ ⁡ ( a , z ) incomplete-Gamma 𝑎 𝑧 {\displaystyle{\displaystyle\Gamma\left(\NVar{a},\NVar{z}\right)}}
    xml-id
    C8.S2.E2.m2adec
    0 references
    Q10771
    Defining formula
    ∫ {\displaystyle{\displaystyle\int}}
    xml-id
    C1.S4.SS4.m3andec
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    Q10811
    Defining formula
    ℜ ⁡ absent {\displaystyle{\displaystyle\Re}}
    xml-id
    C1.S9.E2.m1aadec
    0 references
    Q11400
    Defining formula
    p 𝑝 {\displaystyle{\displaystyle p}}
    xml-id
    C9.S10.XMD2.m1bdec
    0 references
     
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