DLMF:7.14.E1 (Q2433)

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DLMF:7.14.E1
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    Statements

    Defining formula
    ∫ 0 ∞ e 2 ⁢ i ⁢ a ⁢ t ⁢ erfc ⁡ ( b ⁢ t ) ⁢ d t = 1 a ⁢ π ⁢ F ⁡ ( a b ) + i 2 ⁢ a ⁢ ( 1 - e - ( a / b ) 2 ) , superscript subscript 0 superscript 𝑒 2 𝑖 𝑎 𝑡 complementary-error-function 𝑏 𝑡 𝑡 1 𝑎 𝜋 Dawsons-integral 𝑎 𝑏 𝑖 2 𝑎 1 superscript 𝑒 superscript 𝑎 𝑏 2 {\displaystyle{\displaystyle\int_{0}^{\infty}e^{2iat}\operatorname{erfc}\left(% bt\right)\mathrm{d}t={\frac{1}{a\sqrt{\pi}}F\left(\frac{a}{b}\right)+\frac{i}{% 2a}\left(1-e^{-(a/b)^{2}}\right)},}}
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    DLMF id
    DLMF:7.14.E1
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    constraint
    a ∈ ℂ 𝑎 {\displaystyle{\displaystyle a\in\mathbb{C}}}
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    | ph ⁡ b | < 1 4 ⁢ π phase 𝑏 1 4 𝜋 {\displaystyle{\displaystyle|\operatorname{ph}b|<\tfrac{1}{4}\pi}}
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    Symbols used
    Dawson’s integral
    Defining formula
    F ⁡ ( z ) Dawsons-integral 𝑧 {\displaystyle{\displaystyle F\left(\NVar{z}\right)}}
    xml-id
    C7.S2.E5.m2adec
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    the ratio of the circumference of a circle to its diameter
    Defining formula
    π {\displaystyle{\displaystyle\pi}}
    xml-id
    C3.S12.E1.m2adec
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    complementary error function
    Defining formula
    erfc ⁡ z complementary-error-function 𝑧 {\displaystyle{\displaystyle\operatorname{erfc}\NVar{z}}}
    xml-id
    C7.S2.E2.m2adec
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    Q10923
    Defining formula
    ℂ {\displaystyle{\displaystyle\mathbb{C}}}
    xml-id
    introduction.Sx4.p1.t1.r1.m2adec
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    Q10770
    Defining formula
    d x 𝑥 {\displaystyle{\displaystyle\mathrm{d}\NVar{x}}}
    xml-id
    C1.S4.SS4.m1adec
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    Q10784
    Defining formula
    ∈ {\displaystyle{\displaystyle\in}}
    xml-id
    introduction.Sx4.p1.t1.r10.m2adec
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    base of natural logarithm
    Defining formula
    e {\displaystyle{\displaystyle\mathrm{e}}}
    xml-id
    C4.S2.E11.m2adec
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    imaginary unit
    Defining formula
    i imaginary-unit {\displaystyle{\displaystyle\mathrm{i}}}
    xml-id
    C1.S9.E1.m2adec
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    Q10771
    Defining formula
    ∫ {\displaystyle{\displaystyle\int}}
    xml-id
    C1.S4.SS4.m3adec
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    Q10817
    Defining formula
    ph phase {\displaystyle{\displaystyle\operatorname{ph}}}
    xml-id
    C1.S9.E7.m1adec
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