DLMF:9.10.E16 (Q2899): Difference between revisions

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‎Changed an Item: Add constraint
‎Changed an Item: Add constraint
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.m1abdec

Revision as of 14:46, 2 January 2020

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DLMF:9.10.E16
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    Statements

    Defining formula
    ∫ 0 ∞ e - p ⁢ t ⁢ Bi ⁡ ( - t ) ⁢ d t = 1 3 ⁢ e p 3 / 3 ⁢ ( Γ ⁡ ( 2 3 , 1 3 ⁢ p 3 ) Γ ⁡ ( 2 3 ) - Γ ⁡ ( 1 3 , 1 3 ⁢ p 3 ) Γ ⁡ ( 1 3 ) ) , superscript subscript 0 superscript 𝑒 𝑝 𝑡 Airy-Bi 𝑡 𝑡 1 3 superscript 𝑒 superscript 𝑝 3 3 incomplete-Gamma 2 3 1 3 superscript 𝑝 3 Euler-Gamma 2 3 incomplete-Gamma 1 3 1 3 superscript 𝑝 3 Euler-Gamma 1 3 {\displaystyle{\displaystyle\int_{0}^{\infty}e^{-pt}\mathrm{Bi}\left(-t\right)% \mathrm{d}t={\frac{1}{\sqrt{3}}e^{p^{3}/3}\left(\frac{\Gamma\left(\tfrac{2}{3}% ,\tfrac{1}{3}p^{3}\right)}{\Gamma\left(\tfrac{2}{3}\right)}-\frac{\Gamma\left(% \tfrac{1}{3},\tfrac{1}{3}p^{3}\right)}{\Gamma\left(\tfrac{1}{3}\right)}\right)% },}}
    0 references
    DLMF id
    DLMF:9.10.E16
    0 references
    constraint
    ℜ ⁡ p > 0 𝑝 0 {\displaystyle{\displaystyle\Re p>0}}
    0 references
    Symbols used
    Q11081
    Defining formula
    Bi ⁡ ( z ) Airy-Bi 𝑧 {\displaystyle{\displaystyle\mathrm{Bi}\left(\NVar{z}\right)}}
    xml-id
    C9.S2.SS1.m2addec
    0 references
    gamma function
    Defining formula
    Γ ⁡ ( z ) Euler-Gamma 𝑧 {\displaystyle{\displaystyle\Gamma\left(\NVar{z}\right)}}
    xml-id
    C5.S2.E1.m2abdec
    0 references
    Q10770
    Defining formula
    d x 𝑥 {\displaystyle{\displaystyle\mathrm{d}\NVar{x}}}
    xml-id
    C1.S4.SS4.m1aodec
    0 references
    base of natural logarithm
    Defining formula
    e {\displaystyle{\displaystyle\mathrm{e}}}
    xml-id
    C4.S2.E11.m2acdec
    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.m2aadec
    0 references
    Q10771
    Defining formula
    ∫ {\displaystyle{\displaystyle\int}}
    xml-id
    C1.S4.SS4.m3aodec
    0 references
    Q10811
    Defining formula
    ℜ ⁡ absent {\displaystyle{\displaystyle\Re}}
    xml-id
    C1.S9.E2.m1abdec
    0 references
     
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