DLMF:8.20.E3 (Q2716): Difference between revisions

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Property / Symbols used
 
Property / Symbols used: Q11330 / rank
 
Normal rank
Property / Symbols used: Q11330 / qualifier
 
Defining formula:

k 𝑘 {\displaystyle{\displaystyle k}}

k
Property / Symbols used: Q11330 / qualifier
 
xml-id: C8.S1.XMD5.m1bdec

Revision as of 14:14, 2 January 2020

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DLMF:8.20.E3
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    Statements

    Defining formula
    E p ⁡ ( z ) ∼ ± 2 ⁢ π ⁢ i Γ ⁡ ( p ) ⁢ e ∓ p ⁢ π ⁢ i ⁢ z p - 1 + e - z z ⁢ ∑ k = 0 ∞ ( - 1 ) k ⁢ ( p ) k z k , asymptotic-expansion exponential-integral-En 𝑝 𝑧 plus-or-minus 2 𝜋 𝑖 Euler-Gamma 𝑝 superscript 𝑒 minus-or-plus 𝑝 𝜋 𝑖 superscript 𝑧 𝑝 1 superscript 𝑒 𝑧 𝑧 superscript subscript 𝑘 0 superscript 1 𝑘 Pochhammer 𝑝 𝑘 superscript 𝑧 𝑘 {\displaystyle{\displaystyle E_{p}\left(z\right)\sim\pm\frac{2\pi i}{\Gamma% \left(p\right)}e^{\mp p\pi i}z^{p-1}+\frac{e^{-z}}{z}\sum_{k=0}^{\infty}\frac{% (-1)^{k}{\left(p\right)_{k}}}{z^{k}},}}
    0 references
    DLMF id
    DLMF:8.20.E3
    0 references
    constraint
    1 2 ⁢ π + δ ≤ ± ph ⁡ z ≤ 7 2 ⁢ π - δ 1 2 𝜋 𝛿 plus-or-minus phase 𝑧 7 2 𝜋 𝛿 {\displaystyle{\displaystyle\tfrac{1}{2}\pi+\delta\leq\pm\operatorname{ph}z% \leq\tfrac{7}{2}\pi-\delta}}
    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.m2adec
    0 references
    Q10759
    Defining formula
    ( a ) n Pochhammer 𝑎 𝑛 {\displaystyle{\displaystyle{\left(\NVar{a}\right)_{\NVar{n}}}}}
    xml-id
    C5.S2.SS3.m1abdec
    0 references
    Q10931
    Defining formula
    ∼ asymptotic-expansion {\displaystyle{\displaystyle\sim}}
    xml-id
    C2.S1.SS3.p1.m11aadec
    0 references
    the ratio of the circumference of a circle to its diameter
    Defining formula
    π {\displaystyle{\displaystyle\pi}}
    xml-id
    C3.S12.E1.m2aadec
    0 references
    base of natural logarithm
    Defining formula
    e {\displaystyle{\displaystyle\mathrm{e}}}
    xml-id
    C4.S2.E11.m2abdec
    0 references
    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.m2abdec
    0 references
    imaginary unit
    Defining formula
    i imaginary-unit {\displaystyle{\displaystyle\mathrm{i}}}
    xml-id
    C1.S9.E1.m2adec
    0 references
    Q10817
    Defining formula
    ph phase {\displaystyle{\displaystyle\operatorname{ph}}}
    xml-id
    C1.S9.E7.m1aadec
    0 references
    Q11323
    Defining formula
    z 𝑧 {\displaystyle{\displaystyle z}}
    xml-id
    C8.S1.XMD2.m1bdec
    0 references
    Q11377
    Defining formula
    p 𝑝 {\displaystyle{\displaystyle p}}
    xml-id
    C8.S1.XMD4.m1bdec
    0 references
    Q11330
    Defining formula
    k 𝑘 {\displaystyle{\displaystyle k}}
    xml-id
    C8.S1.XMD5.m1bdec
    0 references
     
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