DLMF:13.10.E7 (Q4466)

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DLMF:13.10.E7
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    0 e - z t t b - 1 U ( a , c , t ) d t = Γ ( b ) Γ ( b - c + 1 ) z - b 𝐅 1 2 ( a , b ; a + b - c + 1 ; 1 - 1 z ) , superscript subscript 0 superscript 𝑒 𝑧 𝑡 superscript 𝑡 𝑏 1 Kummer-confluent-hypergeometric-U 𝑎 𝑐 𝑡 𝑡 Euler-Gamma 𝑏 Euler-Gamma 𝑏 𝑐 1 superscript 𝑧 𝑏 hypergeometric-bold-pFq 2 1 𝑎 𝑏 𝑎 𝑏 𝑐 1 1 1 𝑧 {\displaystyle{\displaystyle\int_{0}^{\infty}e^{-zt}t^{b-1}U\left(a,c,t\right)% \mathrm{d}t=\Gamma\left(b\right)\Gamma\left(b-c+1\right)\*z^{-b}{{}_{2}{% \mathbf{F}}_{1}}\left(a,b;a+b-c+1;1-\frac{1}{z}\right),}}
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    DLMF:13.10.E7
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    b > max ( c - 1 , 0 ) 𝑏 𝑐 1 0 {\displaystyle{\displaystyle\Re b>\max\left(\Re c-1,0\right)}}
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    z > 0 𝑧 0 {\displaystyle{\displaystyle\Re z>0}}
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