DLMF:2.3.E8 (Q724)

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DLMF:2.3.E8
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    Statements

    Defining formula
    ∫ 0 ∞ e - x ⁢ t ⁢ q ⁢ ( t ) ⁢ d t ∼ ∑ s = 0 ∞ Γ ⁡ ( s + λ μ ) ⁢ a s x ( s + λ ) / μ , asymptotic-expansion superscript subscript 0 superscript 𝑒 𝑥 𝑡 𝑞 𝑡 𝑡 superscript subscript 𝑠 0 Euler-Gamma 𝑠 𝜆 𝜇 subscript 𝑎 𝑠 superscript 𝑥 𝑠 𝜆 𝜇 {\displaystyle{\displaystyle\int_{0}^{\infty}e^{-xt}q(t)\mathrm{d}t\sim\sum_{s% =0}^{\infty}\Gamma\left(\frac{s+\lambda}{\mu}\right)\frac{a_{s}}{x^{(s+\lambda% )/\mu}},}}
    0 references
    DLMF id
    DLMF:2.3.E8
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    constraint
    x → + ∞ → 𝑥 {\displaystyle{\displaystyle x\to+\infty}}
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    Symbols used
    gamma function
    Defining formula
    Γ ⁡ ( z ) Euler-Gamma 𝑧 {\displaystyle{\displaystyle\Gamma\left(\NVar{z}\right)}}
    xml-id
    C5.S2.E1.m2adec
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    Q10931
    Defining formula
    ∼ asymptotic-expansion {\displaystyle{\displaystyle\sim}}
    xml-id
    C2.S1.SS3.p1.m11addec
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    Q10770
    Defining formula
    d x 𝑥 {\displaystyle{\displaystyle\mathrm{d}\NVar{x}}}
    xml-id
    C1.S4.SS4.m1aedec
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    base of natural logarithm
    Defining formula
    e {\displaystyle{\displaystyle\mathrm{e}}}
    xml-id
    C4.S2.E11.m2addec
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    Q10771
    Defining formula
    ∫ {\displaystyle{\displaystyle\int}}
    xml-id
    C1.S4.SS4.m3aedec
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    Q10949
    Defining formula
    a 𝑎 {\displaystyle{\displaystyle a}}
    xml-id
    C2.S3.XMD12.m1adec
    0 references
    function (locally)
    Defining formula
    q ⁢ ( t ) 𝑞 𝑡 {\displaystyle{\displaystyle q(t)}}
    xml-id
    C2.S3.XMD6.m1adec
    0 references
    Q10950
    Defining formula
    λ 𝜆 {\displaystyle{\displaystyle\lambda}}
    xml-id
    C2.S3.XMD8.m1adec
    0 references
    Q10951
    Defining formula
    μ 𝜇 {\displaystyle{\displaystyle\mu}}
    xml-id
    C2.S3.XMD9.m1adec
    0 references
     
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