DLMF:2.4.E19 (Q774)

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DLMF:2.4.E19
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    Statements

    Defining formula
    I ⁢ ( α , z ) = e - c ⁢ z ⁢ ∫ 𝒬 exp ⁡ ( - z ⁢ ( 1 3 ⁢ w 3 + a ⁢ w 2 + b ⁢ w ) ) ⁢ f ⁢ ( α , w ) ⁢ d w , 𝐼 𝛼 𝑧 superscript 𝑒 𝑐 𝑧 subscript 𝒬 𝑧 1 3 superscript 𝑤 3 𝑎 superscript 𝑤 2 𝑏 𝑤 𝑓 𝛼 𝑤 𝑤 {\displaystyle{\displaystyle I(\alpha,z)=e^{-cz}\int_{\mathscr{Q}}\exp\left(-z% \left(\tfrac{1}{3}w^{3}+aw^{2}+bw\right)\right)f(\alpha,w)\mathrm{d}w,}}
    0 references
    DLMF id
    DLMF:2.4.E19
    0 references
    Symbols used
    Q10770
    Defining formula
    d x 𝑥 {\displaystyle{\displaystyle\mathrm{d}\NVar{x}}}
    xml-id
    C1.S4.SS4.m1ajdec
    0 references
    exponential function
    Defining formula
    exp ⁡ z 𝑧 {\displaystyle{\displaystyle\exp\NVar{z}}}
    xml-id
    C4.S2.E19.m2adec
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    base of natural logarithm
    Defining formula
    e {\displaystyle{\displaystyle\mathrm{e}}}
    xml-id
    C4.S2.E11.m2amdec
    0 references
    Q10771
    Defining formula
    ∫ {\displaystyle{\displaystyle\int}}
    xml-id
    C1.S4.SS4.m3ajdec
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    Q10986
    Defining formula
    I ⁢ ( α , z ) 𝐼 𝛼 𝑧 {\displaystyle{\displaystyle I(\alpha,z)}}
    xml-id
    C2.S4.XMD20.m1adec
    0 references
    Q10987
    Defining formula
    w 𝑤 {\displaystyle{\displaystyle w}}
    xml-id
    C2.S4.XMD21.m1adec
    0 references
    Q10988
    Defining formula
    a 𝑎 {\displaystyle{\displaystyle a}}
    xml-id
    C2.S4.XMD22.m1adec
    0 references
    Q10989
    Defining formula
    b 𝑏 {\displaystyle{\displaystyle b}}
    xml-id
    C2.S4.XMD23.m1adec
    0 references
    Q10990
    Defining formula
    𝒬 𝒬 {\displaystyle{\displaystyle\mathscr{Q}}}
    xml-id
    C2.S4.XMD25.m1dec
    0 references
    Q10991
    Defining formula
    f ⁢ ( α , w ) 𝑓 𝛼 𝑤 {\displaystyle{\displaystyle f(\alpha,w)}}
    xml-id
    C2.S4.XMD26.m1dec
    0 references
     
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