DLMF:13.10.E12 (Q4471)

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DLMF:13.10.E12
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    Statements

    Defining formula
    ∫ 0 ∞ cos ⁡ ( 2 ⁢ x ⁢ t ) ⁢ 𝐌 ⁡ ( a , b , - t 2 ) ⁢ d t = π 2 ⁢ Γ ⁡ ( a ) ⁢ x 2 ⁢ a - 1 ⁢ e - x 2 ⁢ U ⁡ ( b - 1 2 , a + 1 2 , x 2 ) , superscript subscript 0 2 𝑥 𝑡 Kummer-confluent-hypergeometric-bold-M 𝑎 𝑏 superscript 𝑡 2 𝑡 𝜋 2 Euler-Gamma 𝑎 superscript 𝑥 2 𝑎 1 superscript 𝑒 superscript 𝑥 2 Kummer-confluent-hypergeometric-U 𝑏 1 2 𝑎 1 2 superscript 𝑥 2 {\displaystyle{\displaystyle\int_{0}^{\infty}\cos\left(2xt\right){\mathbf{M}}% \left(a,b,-t^{2}\right)\mathrm{d}t=\frac{\sqrt{\pi}}{2\Gamma\left(a\right)}x^{% 2a-1}e^{-x^{2}}U\left(b-\tfrac{1}{2},a+\tfrac{1}{2},x^{2}\right),}}
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    DLMF id
    DLMF:13.10.E12
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    constraint
    ℜ ⁡ a > 0 𝑎 0 {\displaystyle{\displaystyle\Re a>0}}
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