DLMF:13.10.E14 (Q4473)

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DLMF:13.10.E14
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    Statements

    Defining formula
    ∫ 0 ∞ e - t ⁢ t 1 2 ⁢ ν ⁢ 𝐌 ⁡ ( a , b , t ) ⁢ J ν ⁡ ( 2 ⁢ x ⁢ t ) ⁢ d t = x 1 2 ⁢ ν ⁢ e - x Γ ⁡ ( b - a ) ⁢ U ⁡ ( a , a - b + ν + 2 , x ) , superscript subscript 0 superscript 𝑒 𝑡 superscript 𝑡 1 2 𝜈 Kummer-confluent-hypergeometric-bold-M 𝑎 𝑏 𝑡 Bessel-J 𝜈 2 𝑥 𝑡 𝑡 superscript 𝑥 1 2 𝜈 superscript 𝑒 𝑥 Euler-Gamma 𝑏 𝑎 Kummer-confluent-hypergeometric-U 𝑎 𝑎 𝑏 𝜈 2 𝑥 {\displaystyle{\displaystyle\int_{0}^{\infty}e^{-t}t^{\frac{1}{2}\nu}{\mathbf{% M}}\left(a,b,t\right)J_{\nu}\left(2\sqrt{xt}\right)\mathrm{d}t=\frac{x^{\frac{% 1}{2}\nu}e^{-x}}{\Gamma\left(b-a\right)}U\left(a,a-b+\nu+2,x\right),}}
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    DLMF id
    DLMF:13.10.E14
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    constraint
    - 1 < ℜ ⁡ ν < 2 ⁢ ℜ ⁡ ( b - a ) - 1 2 1 𝜈 2 𝑏 𝑎 1 2 {\displaystyle{\displaystyle-1<\Re\nu<2\Re(b-a)-\tfrac{1}{2}}}
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    x > 0 𝑥 0 {\displaystyle{\displaystyle x>0}}
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    - 1 < ℜ ⁡ ν < 2 ⁢ ℜ ⁡ ( b - a ) - 1 2 1 𝜈 2 𝑏 𝑎 1 2 {\displaystyle{\displaystyle-1<\Re\nu<2\Re(b-a)-\tfrac{1}{2}}}
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    Symbols used
    Bessel function of the first kind
    Defining formula
    J ν ⁡ ( z ) Bessel-J 𝜈 𝑧 {\displaystyle{\displaystyle J_{\NVar{\nu}}\left(\NVar{z}\right)}}
    xml-id
    C10.S2.E2.m2aadec
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    gamma function
    Defining formula
    Γ ⁡ ( z ) Euler-Gamma 𝑧 {\displaystyle{\displaystyle\Gamma\left(\NVar{z}\right)}}
    xml-id
    C5.S2.E1.m2aidec
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    Kummer confluent hypergeometric function
    Defining formula
    U ⁡ ( a , b , z ) Kummer-confluent-hypergeometric-U 𝑎 𝑏 𝑧 {\displaystyle{\displaystyle U\left(\NVar{a},\NVar{b},\NVar{z}\right)}}
    xml-id
    C13.S2.E6.m2afdec
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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.m2aidec
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    Q10770
    Defining formula
    d x 𝑥 {\displaystyle{\displaystyle\mathrm{d}\NVar{x}}}
    xml-id
    C1.S4.SS4.m1amdec
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    base of natural logarithm
    Defining formula
    e {\displaystyle{\displaystyle\mathrm{e}}}
    xml-id
    C4.S2.E11.m2aidec
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    Q10771
    Defining formula
    ∫ {\displaystyle{\displaystyle\int}}
    xml-id
    C1.S4.SS4.m3amdec
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    Q11566
    Defining formula
    x 𝑥 {\displaystyle{\displaystyle x}}
    xml-id
    C13.S1.XMD4.m1bdec
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    Q10811
    Defining formula
    ℜ ⁡ absent {\displaystyle{\displaystyle\Re}}
    xml-id
    C1.S9.E2.m1akdec
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