DLMF:6.7.E5 (Q2258): Difference between revisions

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Property / Symbols used
 
Property / Symbols used: Q11307 / rank
 
Normal rank
Property / Symbols used: Q11307 / qualifier
 
Defining formula:

x 𝑥 {\displaystyle{\displaystyle x}}

x
Property / Symbols used: Q11307 / qualifier
 
xml-id: C6.S1.XMD1.m1cdec

Latest revision as of 01:26, 2 January 2020

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DLMF:6.7.E5
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    Statements

    Defining formula
    ∫ x ∞ e - t a 2 + t 2 ⁢ d t = - 1 2 ⁢ a ⁢ i ⁢ ( e i ⁢ a ⁢ E 1 ⁡ ( x + i ⁢ a ) - e - i ⁢ a ⁢ E 1 ⁡ ( x - i ⁢ a ) ) , superscript subscript 𝑥 superscript 𝑒 𝑡 superscript 𝑎 2 superscript 𝑡 2 𝑡 1 2 𝑎 𝑖 superscript 𝑒 𝑖 𝑎 exponential-integral 𝑥 𝑖 𝑎 superscript 𝑒 𝑖 𝑎 exponential-integral 𝑥 𝑖 𝑎 {\displaystyle{\displaystyle\int_{x}^{\infty}\frac{e^{-t}}{a^{2}+t^{2}}\mathrm% {d}t=-\frac{1}{2ai}\left(e^{ia}E_{1}\left(x+ia\right)-e^{-ia}E_{1}\left(x-ia% \right)\right),}}
    0 references
    DLMF id
    DLMF:6.7.E5
    0 references
    constraint
    x ∈ ℝ 𝑥 {\displaystyle{\displaystyle x\in\mathbb{R}}}
    0 references
    a > 0 𝑎 0 {\displaystyle{\displaystyle a>0}}
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    x ∈ ℝ 𝑥 ℝ {\displaystyle{\displaystyle x\in\mathbb{R}}}
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    Symbols used
    Q10770
    Defining formula
    d x 𝑥 {\displaystyle{\displaystyle\mathrm{d}\NVar{x}}}
    xml-id
    C1.S4.SS4.m1addec
    0 references
    Q10784
    Defining formula
    ∈ {\displaystyle{\displaystyle\in}}
    xml-id
    introduction.Sx4.p1.t1.r10.m2adec
    0 references
    base of natural logarithm
    Defining formula
    e {\displaystyle{\displaystyle\mathrm{e}}}
    xml-id
    C4.S2.E11.m2addec
    0 references
    exponential integral
    Defining formula
    E 1 ⁡ ( z ) exponential-integral 𝑧 {\displaystyle{\displaystyle E_{1}\left(\NVar{z}\right)}}
    xml-id
    C6.S2.E1.m2acdec
    0 references
    imaginary unit
    Defining formula
    i imaginary-unit {\displaystyle{\displaystyle\mathrm{i}}}
    xml-id
    C1.S9.E1.m2acdec
    0 references
    Q10771
    Defining formula
    ∫ {\displaystyle{\displaystyle\int}}
    xml-id
    C1.S4.SS4.m3addec
    0 references
    Q10812
    Defining formula
    ℝ {\displaystyle{\displaystyle\mathbb{R}}}
    xml-id
    introduction.Sx4.p2.t1.r14.m2adec
    0 references
    Q11307
    Defining formula
    x 𝑥 {\displaystyle{\displaystyle x}}
    xml-id
    C6.S1.XMD1.m1cdec
    0 references
     
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