DLMF:8.19.E18 (Q2704)

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DLMF:8.19.E18
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    Statements

    test
    E p ⁡ ( z ⁢ e 2 ⁢ m ⁢ π ⁢ i ) = 2 ⁢ π ⁢ i ⁢ e m ⁢ p ⁢ π ⁢ i Γ ⁡ ( p ) ⁢ sin ⁡ ( m ⁢ p ⁢ π ) sin ⁡ ( p ⁢ π ) ⁢ z p - 1 + E p ⁡ ( z ) , exponential-integral-En 𝑝 𝑧 superscript 𝑒 2 𝑚 𝜋 𝑖 2 𝜋 𝑖 superscript 𝑒 𝑚 𝑝 𝜋 𝑖 Euler-Gamma 𝑝 𝑚 𝑝 𝜋 𝑝 𝜋 superscript 𝑧 𝑝 1 exponential-integral-En 𝑝 𝑧 {\displaystyle{\displaystyle E_{p}\left(ze^{2m\pi i}\right)=\frac{2\pi ie^{mp% \pi i}}{\Gamma\left(p\right)}\frac{\sin\left(mp\pi\right)}{\sin\left(p\pi% \right)}z^{p-1}+E_{p}\left(z\right),}}
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    DLMF id
    DLMF:8.19.E18
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    constraint
    m ∈ ℤ 𝑚 {\displaystyle{\displaystyle m\in\mathbb{Z}}}
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    m ∈ ℤ 𝑚 ℤ {\displaystyle{\displaystyle m\in\mathbb{Z}}}
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    z ≠ 0 𝑧 0 {\displaystyle{\displaystyle z\neq 0}}
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    Symbols used
    gamma function
    test
    Γ ⁡ ( z ) Euler-Gamma 𝑧 {\displaystyle{\displaystyle\Gamma\left(\NVar{z}\right)}}
    xml-id
    C5.S2.E1.m2addec
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    the ratio of the circumference of a circle to its diameter
    test
    π {\displaystyle{\displaystyle\pi}}
    xml-id
    C3.S12.E1.m2acdec
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    Q10784
    test
    ∈ {\displaystyle{\displaystyle\in}}
    xml-id
    introduction.Sx4.p1.t1.r10.m2adec
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    base of natural logarithm
    test
    e {\displaystyle{\displaystyle\mathrm{e}}}
    xml-id
    C4.S2.E11.m2aldec
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    generalized exponential integral
    test
    E p ⁡ ( z ) exponential-integral-En 𝑝 𝑧 {\displaystyle{\displaystyle E_{\NVar{p}}\left(\NVar{z}\right)}}
    xml-id
    C8.S19.E1.m2avdec
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    imaginary unit
    test
    i imaginary-unit {\displaystyle{\displaystyle\mathrm{i}}}
    xml-id
    C1.S9.E1.m2addec
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    Q10818
    test
    ℤ {\displaystyle{\displaystyle\mathbb{Z}}}
    xml-id
    introduction.Sx4.p2.t1.r20.m2adec
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    sine function
    test
    sin ⁡ z 𝑧 {\displaystyle{\displaystyle\sin\NVar{z}}}
    xml-id
    C4.S14.E1.m2adec
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    Q11323
    test
    z 𝑧 {\displaystyle{\displaystyle z}}
    xml-id
    C8.S1.XMD2.m1pdec
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    Q11377
    test
    p 𝑝 {\displaystyle{\displaystyle p}}
    xml-id
    C8.S1.XMD4.m1ndec
    0 references
     
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