DLMF:7.14.E8 (Q2440)

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DLMF:7.14.E8
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    Statements

    test
    ∫ 0 ∞ e - a ⁢ t ⁢ S ⁡ ( 2 ⁢ t π ) ⁢ d t = ( a 2 + 1 - a ) 1 2 2 ⁢ a ⁢ a 2 + 1 , superscript subscript 0 superscript 𝑒 𝑎 𝑡 Fresnel-sine-integral 2 𝑡 𝜋 𝑡 superscript superscript 𝑎 2 1 𝑎 1 2 2 𝑎 superscript 𝑎 2 1 {\displaystyle{\displaystyle\int_{0}^{\infty}e^{-at}S\left(\sqrt{\frac{2t}{\pi% }}\right)\mathrm{d}t=\frac{(\sqrt{a^{2}+1}-a)^{\frac{1}{2}}}{2a\sqrt{a^{2}+1}}% ,}}
    0 references
    DLMF id
    DLMF:7.14.E8
    0 references
    constraint
    ℜ ⁡ a > 0 𝑎 0 {\displaystyle{\displaystyle\Re a>0}}
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    Symbols used
    Fresnel integral
    test
    S ⁡ ( z ) Fresnel-sine-integral 𝑧 {\displaystyle{\displaystyle S\left(\NVar{z}\right)}}
    xml-id
    C7.S2.E8.m2aadec
    0 references
     
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