DLMF:9.10.E19 (Q2903)

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DLMF:9.10.E19
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    Statements

    Defining formula
    Bi ⁡ ( x ) = 3 ⁢ x 5 / 4 ⁢ e ( 2 / 3 ) ⁢ x 3 / 2 2 ⁢ π ⁢ ⨍ 0 ∞ t - 3 / 4 ⁢ e - ( 2 / 3 ) ⁢ t 3 / 2 ⁢ Ai ⁡ ( t ) x 3 / 2 - t 3 / 2 ⁢ d t , Airy-Bi 𝑥 3 superscript 𝑥 5 4 superscript 𝑒 2 3 superscript 𝑥 3 2 2 𝜋 superscript subscript principal-value-integral 0 superscript 𝑡 3 4 superscript 𝑒 2 3 superscript 𝑡 3 2 Airy-Ai 𝑡 superscript 𝑥 3 2 superscript 𝑡 3 2 𝑡 {\displaystyle{\displaystyle\mathrm{Bi}\left(x\right)=\frac{3x^{5/4}e^{(2/3)x^% {3/2}}}{2\pi}\*\pvint_{0}^{\infty}\frac{t^{-3/4}e^{-(2/3)t^{3/2}}\mathrm{Ai}% \left(t\right)}{x^{3/2}-t^{3/2}}\mathrm{d}t,}}
    0 references
    DLMF id
    DLMF:9.10.E19
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    constraint
    x > 0 𝑥 0 {\displaystyle{\displaystyle x>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.m1ajdec
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    Q11081
    Defining formula
    Bi ⁡ ( z ) Airy-Bi 𝑧 {\displaystyle{\displaystyle\mathrm{Bi}\left(\NVar{z}\right)}}
    xml-id
    C9.S2.SS1.m2aedec
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    the ratio of the circumference of a circle to its diameter
    Defining formula
    π {\displaystyle{\displaystyle\pi}}
    xml-id
    C3.S12.E1.m2ahdec
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    Q10770
    Defining formula
    d x 𝑥 {\displaystyle{\displaystyle\mathrm{d}\NVar{x}}}
    xml-id
    C1.S4.SS4.m1ardec
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    base of natural logarithm
    Defining formula
    e {\displaystyle{\displaystyle\mathrm{e}}}
    xml-id
    C4.S2.E11.m2aedec
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    Cauchy principal value
    Defining formula
    ⨍ a b superscript subscript principal-value-integral 𝑎 𝑏 {\displaystyle{\displaystyle\pvint_{\NVar{a}}^{\NVar{b}}}}
    xml-id
    C1.S4.E24.m2adec
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    Q11382
    Defining formula
    x 𝑥 {\displaystyle{\displaystyle x}}
    xml-id
    C9.S1.XMD2.m1ddec
    0 references
     
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