DLMF:13.10.E9 (Q4468)

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DLMF:13.10.E9
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    Statements

    Defining formula
    1 2 ⁢ π ⁢ i ⁢ ∫ - ∞ ( 0 + ) e t ⁢ z ⁢ t - a ⁢ U ⁡ ( a , b , y / t ) ⁢ d t = 2 ⁢ z 1 2 ⁢ ( 2 ⁢ a - b - 1 ) ⁢ y 1 2 ⁢ ( 1 - b ) Γ ⁡ ( a ) ⁢ Γ ⁡ ( a - b + 1 ) ⁢ K b - 1 ⁡ ( 2 ⁢ z ⁢ y ) , 1 2 𝜋 imaginary-unit superscript subscript limit-from 0 superscript 𝑒 𝑡 𝑧 superscript 𝑡 𝑎 Kummer-confluent-hypergeometric-U 𝑎 𝑏 𝑦 𝑡 𝑡 2 superscript 𝑧 1 2 2 𝑎 𝑏 1 superscript 𝑦 1 2 1 𝑏 Euler-Gamma 𝑎 Euler-Gamma 𝑎 𝑏 1 modified-Bessel-second-kind 𝑏 1 2 𝑧 𝑦 {\displaystyle{\displaystyle\frac{1}{2\pi\mathrm{i}}\int_{-\infty}^{(0+)}e^{tz% }t^{-a}U\left(a,b,\ifrac{y}{t}\right)\mathrm{d}t=\frac{2z^{\frac{1}{2}(2a-b-1)% }y^{\frac{1}{2}(1-b)}}{\Gamma\left(a\right)\Gamma\left(a-b+1\right)}K_{b-1}% \left(2\sqrt{zy}\right),}}
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    DLMF id
    DLMF:13.10.E9
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    constraint
    ℜ ⁡ z > 0 𝑧 0 {\displaystyle{\displaystyle\Re z>0}}
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