DLMF:6.7.E3 (Q2256)

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DLMF:6.7.E3
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    Statements

    Defining formula
    ∫ x ∞ e i ⁢ t a 2 + t 2 ⁢ d t = i 2 ⁢ a ⁢ ( e a ⁢ E 1 ⁡ ( a - i ⁢ x ) - e - a ⁢ E 1 ⁡ ( - a - i ⁢ x ) ) , superscript subscript 𝑥 superscript 𝑒 𝑖 𝑡 superscript 𝑎 2 superscript 𝑡 2 𝑡 𝑖 2 𝑎 superscript 𝑒 𝑎 exponential-integral 𝑎 𝑖 𝑥 superscript 𝑒 𝑎 exponential-integral 𝑎 𝑖 𝑥 {\displaystyle{\displaystyle\int_{x}^{\infty}\frac{e^{it}}{a^{2}+t^{2}}\mathrm% {d}t=\frac{i}{2a}\left(e^{a}E_{1}\left(a-ix\right)-e^{-a}E_{1}\left(-a-ix% \right)\right),}}
    0 references
    DLMF id
    DLMF:6.7.E3
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    constraint
    x > 0 𝑥 0 {\displaystyle{\displaystyle x>0}}
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    Symbols used
    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.m2abdec
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    exponential integral
    Defining formula
    E 1 ⁡ ( z ) exponential-integral 𝑧 {\displaystyle{\displaystyle E_{1}\left(\NVar{z}\right)}}
    xml-id
    C6.S2.E1.m2aadec
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    imaginary unit
    Defining formula
    i imaginary-unit {\displaystyle{\displaystyle\mathrm{i}}}
    xml-id
    C1.S9.E1.m2aadec
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    Q10771
    Defining formula
    ∫ {\displaystyle{\displaystyle\int}}
    xml-id
    C1.S4.SS4.m3abdec
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    Q11307
    Defining formula
    x 𝑥 {\displaystyle{\displaystyle x}}
    xml-id
    C6.S1.XMD1.m1adec
    0 references
     
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