DLMF:13.10.E6 (Q4465)

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DLMF:13.10.E6
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    Statements

    Defining formula
    ∫ 0 ∞ e - z ⁢ t - t 2 ⁢ t 2 ⁢ b - 2 ⁢ 𝐌 ⁡ ( a , b , t 2 ) ⁢ d t = 1 2 ⁢ π - 1 2 ⁢ Γ ⁡ ( b - 1 2 ) ⁢ U ⁡ ( b - 1 2 , a + 1 2 , 1 4 ⁢ z 2 ) , superscript subscript 0 superscript 𝑒 𝑧 𝑡 superscript 𝑡 2 superscript 𝑡 2 𝑏 2 Kummer-confluent-hypergeometric-bold-M 𝑎 𝑏 superscript 𝑡 2 𝑡 1 2 superscript 𝜋 1 2 Euler-Gamma 𝑏 1 2 Kummer-confluent-hypergeometric-U 𝑏 1 2 𝑎 1 2 1 4 superscript 𝑧 2 {\displaystyle{\displaystyle\int_{0}^{\infty}e^{-zt-t^{2}}t^{2b-2}{\mathbf{M}}% \left(a,b,t^{2}\right)\mathrm{d}t=\tfrac{1}{2}\pi^{-\frac{1}{2}}\Gamma\left(b-% \tfrac{1}{2}\right)U\left(b-\tfrac{1}{2},a+\tfrac{1}{2},\tfrac{1}{4}z^{2}% \right),}}
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    DLMF id
    DLMF:13.10.E6
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    constraint
    ℜ ⁡ b > 1 2 𝑏 1 2 {\displaystyle{\displaystyle\Re b>\tfrac{1}{2}}}
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    ℜ ⁡ z > 0 𝑧 0 {\displaystyle{\displaystyle\Re z>0}}
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