DLMF:2.4.E12 (Q766)

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DLMF:2.4.E12
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    Statements

    Defining formula
    I ⁢ ( z ) ∼ e - z ⁢ p ⁢ ( a ) ⁢ ∑ s = 0 ∞ Γ ⁡ ( s + λ μ ) ⁢ b s z ( s + λ ) / μ asymptotic-expansion 𝐼 𝑧 superscript 𝑒 𝑧 𝑝 𝑎 superscript subscript 𝑠 0 Euler-Gamma 𝑠 𝜆 𝜇 subscript 𝑏 𝑠 superscript 𝑧 𝑠 𝜆 𝜇 {\displaystyle{\displaystyle I(z)\sim e^{-zp(a)}\sum_{s=0}^{\infty}\Gamma\left% (\frac{s+\lambda}{\mu}\right)\frac{b_{s}}{z^{(s+\lambda)/\mu}}}}
    0 references
    DLMF id
    DLMF:2.4.E12
    0 references
    Symbols used
    gamma function
    Defining formula
    Γ ⁡ ( z ) Euler-Gamma 𝑧 {\displaystyle{\displaystyle\Gamma\left(\NVar{z}\right)}}
    xml-id
    C5.S2.E1.m2abdec
    0 references
    Q10931
    Defining formula
    ∼ asymptotic-expansion {\displaystyle{\displaystyle\sim}}
    xml-id
    C2.S1.SS3.p1.m11abdec
    0 references
    base of natural logarithm
    Defining formula
    e {\displaystyle{\displaystyle\mathrm{e}}}
    xml-id
    C4.S2.E11.m2aidec
    0 references
    Q10971
    Defining formula
    λ 𝜆 {\displaystyle{\displaystyle\lambda}}
    xml-id
    C2.S4.XMD1.m1cdec
    0 references
    Q10972
    Defining formula
    a 𝑎 {\displaystyle{\displaystyle a}}
    xml-id
    C2.S4.XMD11.m1ddec
    0 references
    Q10980
    Defining formula
    b 𝑏 {\displaystyle{\displaystyle b}}
    xml-id
    C2.S4.XMD12.m1adec
    0 references
    contour integral (locally)
    Defining formula
    I ⁢ ( z ) 𝐼 𝑧 {\displaystyle{\displaystyle I(z)}}
    xml-id
    C2.S4.XMD13.m1adec
    0 references
    Q10981
    Defining formula
    p ⁢ ( t ) 𝑝 𝑡 {\displaystyle{\displaystyle p(t)}}
    xml-id
    C2.S4.XMD15.m1bdec
    0 references
    Q10974
    Defining formula
    μ 𝜇 {\displaystyle{\displaystyle\mu}}
    xml-id
    C2.S4.XMD2.m1cdec
    0 references
     
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