DLMF:13.10.E4 (Q4463)

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DLMF:13.10.E4
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    Statements

    Defining formula
    ∫ 0 ∞ e - z ⁢ t ⁢ t b - 1 ⁢ 𝐌 ⁡ ( a , b , t ) ⁢ d t = z - b ⁢ ( 1 - 1 z ) - a , superscript subscript 0 superscript 𝑒 𝑧 𝑡 superscript 𝑡 𝑏 1 Kummer-confluent-hypergeometric-bold-M 𝑎 𝑏 𝑡 𝑡 superscript 𝑧 𝑏 superscript 1 1 𝑧 𝑎 {\displaystyle{\displaystyle\int_{0}^{\infty}e^{-zt}t^{b-1}{\mathbf{M}}\left(a% ,b,t\right)\mathrm{d}t=z^{-b}\left(1-\frac{1}{z}\right)^{-a},}}
    0 references
    DLMF id
    DLMF:13.10.E4
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    constraint
    ℜ ⁡ b > 0 𝑏 0 {\displaystyle{\displaystyle\Re b>0}}
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    ℜ ⁡ z > 1 𝑧 1 {\displaystyle{\displaystyle\Re z>1}}
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    Symbols used
    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.m2abdec
    0 references
    Q10770
    Defining formula
    d x 𝑥 {\displaystyle{\displaystyle\mathrm{d}\NVar{x}}}
    xml-id
    C1.S4.SS4.m1acdec
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    base of natural logarithm
    Defining formula
    e {\displaystyle{\displaystyle\mathrm{e}}}
    xml-id
    C4.S2.E11.m2aadec
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    Q10771
    Defining formula
    ∫ {\displaystyle{\displaystyle\int}}
    xml-id
    C1.S4.SS4.m3acdec
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