DLMF:2.11.E11 (Q1036)

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DLMF:2.11.E11
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    Statements

    Defining formula
    F n + p ⁡ ( z ) = e - z 2 ⁢ π ⁢ ∫ 0 ∞ e - z ⁢ t ⁢ t n + p - 1 1 + t ⁢ d t = Γ ⁡ ( n + p ) 2 ⁢ π ⁢ E n + p ⁡ ( z ) z n + p - 1 . terminant-function 𝑛 𝑝 𝑧 superscript 𝑒 𝑧 2 𝜋 superscript subscript 0 superscript 𝑒 𝑧 𝑡 superscript 𝑡 𝑛 𝑝 1 1 𝑡 𝑡 Euler-Gamma 𝑛 𝑝 2 𝜋 exponential-integral-En 𝑛 𝑝 𝑧 superscript 𝑧 𝑛 𝑝 1 {\displaystyle{\displaystyle F_{n+p}\left(z\right)=\frac{e^{-z}}{2\pi}\int_{0}% ^{\infty}\frac{e^{-zt}t^{n+p-1}}{1+t}\mathrm{d}t=\frac{\Gamma\left(n+p\right)}% {2\pi}\frac{E_{n+p}\left(z\right)}{z^{n+p-1}}.}}
    0 references
    DLMF id
    DLMF:2.11.E11
    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.m2acdec
    0 references
    the ratio of the circumference of a circle to its diameter
    Defining formula
    π {\displaystyle{\displaystyle\pi}}
    xml-id
    C3.S12.E1.m2addec
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    Q10770
    Defining formula
    d x 𝑥 {\displaystyle{\displaystyle\mathrm{d}\NVar{x}}}
    xml-id
    C1.S4.SS4.m1abdec
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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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    generalized exponential integral
    Defining formula
    E p ⁡ ( z ) exponential-integral-En 𝑝 𝑧 {\displaystyle{\displaystyle E_{\NVar{p}}\left(\NVar{z}\right)}}
    xml-id
    C8.S19.E1.m2addec
    0 references
    Q10771
    Defining formula
    ∫ {\displaystyle{\displaystyle\int}}
    xml-id
    C1.S4.SS4.m3abdec
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