DLMF:13.10.E3 (Q4462)

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DLMF:13.10.E3
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    Statements

    Defining formula
    ∫ 0 ∞ e - z ⁢ t ⁢ t b - 1 ⁢ 𝐌 ⁡ ( a , c , k ⁢ t ) ⁢ d t = Γ ⁡ ( b ) ⁢ z - b ⁢ 𝐅 1 2 ⁡ ( a , b ; c ; k / z ) , superscript subscript 0 superscript 𝑒 𝑧 𝑡 superscript 𝑡 𝑏 1 Kummer-confluent-hypergeometric-bold-M 𝑎 𝑐 𝑘 𝑡 𝑡 Euler-Gamma 𝑏 superscript 𝑧 𝑏 hypergeometric-bold-pFq 2 1 𝑎 𝑏 𝑐 𝑘 𝑧 {\displaystyle{\displaystyle\int_{0}^{\infty}e^{-zt}t^{b-1}{\mathbf{M}}\left(a% ,c,kt\right)\mathrm{d}t=\Gamma\left(b\right)z^{-b}{{}_{2}{\mathbf{F}}_{1}}% \left(a,b;c;\ifrac{k}{z}\right),}}
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    DLMF id
    DLMF:13.10.E3
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    constraint
    ℜ ⁡ b > 0 𝑏 0 {\displaystyle{\displaystyle\Re b>0}}
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    ℜ ⁡ z > max ⁡ ( ℜ ⁡ k , 0 ) 𝑧 𝑘 0 {\displaystyle{\displaystyle\Re z>\max\left(\Re k,0\right)}}
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    Symbols used
    gamma function
    Defining formula
    Γ ⁡ ( z ) Euler-Gamma 𝑧 {\displaystyle{\displaystyle\Gamma\left(\NVar{z}\right)}}
    xml-id
    C5.S2.E1.m2adec
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    Olver’s confluent hypergeometric function
    Defining formula
    𝐌 ⁡ ( a , b , z ) Kummer-confluent-hypergeometric-bold-M 𝑎 𝑏 𝑧 {\displaystyle{\displaystyle{\mathbf{M}}\left(\NVar{a},\NVar{b},\NVar{z}\right% )}}
    xml-id
    C13.S2.E3.m2aadec
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