DLMF:13.10.E13 (Q4472)

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DLMF:13.10.E13
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    Statements

    test
    ∫ 0 ∞ e - t ⁢ t b - 1 - 1 2 ⁢ ν ⁢ 𝐌 ⁡ ( a , b , t ) ⁢ J ν ⁡ ( 2 ⁢ x ⁢ t ) ⁢ d t = x - a + 1 2 ⁢ ν ⁢ e - x ⁢ 𝐌 ⁡ ( ν - b + 1 , ν - a + 1 , x ) , superscript subscript 0 superscript 𝑒 𝑡 superscript 𝑡 𝑏 1 1 2 𝜈 Kummer-confluent-hypergeometric-bold-M 𝑎 𝑏 𝑡 Bessel-J 𝜈 2 𝑥 𝑡 𝑡 superscript 𝑥 𝑎 1 2 𝜈 superscript 𝑒 𝑥 Kummer-confluent-hypergeometric-bold-M 𝜈 𝑏 1 𝜈 𝑎 1 𝑥 {\displaystyle{\displaystyle\int_{0}^{\infty}e^{-t}t^{b-1-\frac{1}{2}\nu}{% \mathbf{M}}\left(a,b,t\right)J_{\nu}\left(2\sqrt{xt}\right)\mathrm{d}t=x^{-a+% \frac{1}{2}\nu}e^{-x}{\mathbf{M}}\left(\nu-b+1,\nu-a+1,x\right),}}
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    DLMF id
    DLMF:13.10.E13
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    constraint
    2 ⁢ ℜ ⁡ a < ℜ ⁡ ν + 5 2 2 𝑎 𝜈 5 2 {\displaystyle{\displaystyle 2\Re a<\Re\nu+\tfrac{5}{2}}}
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    ℜ ⁡ b > 0 𝑏 0 {\displaystyle{\displaystyle\Re b>0}}
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    x > 0 𝑥 0 {\displaystyle{\displaystyle x>0}}
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    2 ⁢ ℜ ⁡ a < ℜ ⁡ ν + 5 2 2 𝑎 𝜈 5 2 {\displaystyle{\displaystyle 2\Re a<\Re\nu+\tfrac{5}{2}}}
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    ℜ ⁡ b > 0 𝑏 0 {\displaystyle{\displaystyle\Re b>0}}
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    Symbols used
    Bessel function of the first kind
    test
    J ν ⁡ ( z ) Bessel-J 𝜈 𝑧 {\displaystyle{\displaystyle J_{\NVar{\nu}}\left(\NVar{z}\right)}}
    xml-id
    C10.S2.E2.m2adec
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