DLMF:13.10.E9 (Q4468)

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DLMF:13.10.E9
No description defined

    Statements

    test
    1 2 ⁢ π ⁢ i ⁢ ∫ - ∞ ( 0 + ) e t ⁢ z ⁢ t - a ⁢ U ⁡ ( a , b , y / t ) ⁢ d t = 2 ⁢ z 1 2 ⁢ ( 2 ⁢ a - b - 1 ) ⁢ y 1 2 ⁢ ( 1 - b ) Γ ⁡ ( a ) ⁢ Γ ⁡ ( a - b + 1 ) ⁢ K b - 1 ⁡ ( 2 ⁢ z ⁢ y ) , 1 2 𝜋 imaginary-unit superscript subscript limit-from 0 superscript 𝑒 𝑡 𝑧 superscript 𝑡 𝑎 Kummer-confluent-hypergeometric-U 𝑎 𝑏 𝑦 𝑡 𝑡 2 superscript 𝑧 1 2 2 𝑎 𝑏 1 superscript 𝑦 1 2 1 𝑏 Euler-Gamma 𝑎 Euler-Gamma 𝑎 𝑏 1 modified-Bessel-second-kind 𝑏 1 2 𝑧 𝑦 {\displaystyle{\displaystyle\frac{1}{2\pi\mathrm{i}}\int_{-\infty}^{(0+)}e^{tz% }t^{-a}U\left(a,b,\ifrac{y}{t}\right)\mathrm{d}t=\frac{2z^{\frac{1}{2}(2a-b-1)% }y^{\frac{1}{2}(1-b)}}{\Gamma\left(a\right)\Gamma\left(a-b+1\right)}K_{b-1}% \left(2\sqrt{zy}\right),}}
    0 references
    DLMF id
    DLMF:13.10.E9
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    constraint
    ℜ ⁡ z > 0 𝑧 0 {\displaystyle{\displaystyle\Re z>0}}
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    Symbols used
    gamma function
    test
    Γ ⁡ ( z ) Euler-Gamma 𝑧 {\displaystyle{\displaystyle\Gamma\left(\NVar{z}\right)}}
    xml-id
    C5.S2.E1.m2aedec
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    Kummer confluent hypergeometric function
    test
    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.m2acdec
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    Q10770
    test
    d x 𝑥 {\displaystyle{\displaystyle\mathrm{d}\NVar{x}}}
    xml-id
    C1.S4.SS4.m1ahdec
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    base of natural logarithm
    test
    e {\displaystyle{\displaystyle\mathrm{e}}}
    xml-id
    C4.S2.E11.m2afdec
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    imaginary unit
    test
    i imaginary-unit {\displaystyle{\displaystyle\mathrm{i}}}
    xml-id
    C1.S9.E1.m2aadec
    0 references
    Q10771
    test
    ∫ {\displaystyle{\displaystyle\int}}
    xml-id
    C1.S4.SS4.m3ahdec
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    modified Bessel function of the second kind
    test
    K ν ⁡ ( z ) modified-Bessel-second-kind 𝜈 𝑧 {\displaystyle{\displaystyle K_{\NVar{\nu}}\left(\NVar{z}\right)}}
    xml-id
    C10.S25.E3.m2adec
    0 references
    Q10811
    test
    ℜ ⁡ absent {\displaystyle{\displaystyle\Re}}
    xml-id
    C1.S9.E2.m1afdec
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    the ratio of the circumference of a circle to its diameter
    test
    π {\displaystyle{\displaystyle\pi}}
    xml-id
    C3.S12.E1.m2abdec
    0 references
    Q11557
    test
    z 𝑧 {\displaystyle{\displaystyle z}}
    xml-id
    C13.S1.XMD6.m1gdec
    0 references
    Q11580
    test
    y 𝑦 {\displaystyle{\displaystyle y}}
    xml-id
    C13.S1.XMD5.m1adec
    0 references
     
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