DLMF:13.4.E7 (Q4370)

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DLMF:13.4.E7
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    Statements

    Defining formula
    U ⁡ ( a , b , z ) = 2 ⁢ z 1 2 - 1 2 ⁢ b Γ ⁡ ( a ) ⁢ Γ ⁡ ( a - b + 1 ) ⁢ ∫ 0 ∞ e - t ⁢ t a - 1 2 ⁢ b - 1 2 ⁢ K b - 1 ⁡ ( 2 ⁢ z ⁢ t ) ⁢ d t , Kummer-confluent-hypergeometric-U 𝑎 𝑏 𝑧 2 superscript 𝑧 1 2 1 2 𝑏 Euler-Gamma 𝑎 Euler-Gamma 𝑎 𝑏 1 superscript subscript 0 superscript 𝑒 𝑡 superscript 𝑡 𝑎 1 2 𝑏 1 2 modified-Bessel-second-kind 𝑏 1 2 𝑧 𝑡 𝑡 {\displaystyle{\displaystyle U\left(a,b,z\right)=\frac{2z^{\frac{1}{2}-\frac{1% }{2}b}}{\Gamma\left(a\right)\Gamma\left(a-b+1\right)}\*\int_{0}^{\infty}e^{-t}% t^{a-\frac{1}{2}b-\frac{1}{2}}K_{b-1}\left(2\sqrt{zt}\right)\mathrm{d}t,}}
    0 references
    DLMF id
    DLMF:13.4.E7
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    constraint
    ℜ ⁡ a > max ⁡ ( ℜ ⁡ b - 1 , 0 ) 𝑎 𝑏 1 0 {\displaystyle{\displaystyle\Re a>\max\left(\Re b-1,0\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.m2afdec
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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.m2acdec
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    Q10770
    Defining formula
    d x 𝑥 {\displaystyle{\displaystyle\mathrm{d}\NVar{x}}}
    xml-id
    C1.S4.SS4.m1afdec
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    base of natural logarithm
    Defining formula
    e {\displaystyle{\displaystyle\mathrm{e}}}
    xml-id
    C4.S2.E11.m2aedec
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    Q10771
    Defining formula
    ∫ {\displaystyle{\displaystyle\int}}
    xml-id
    C1.S4.SS4.m3afdec
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    modified Bessel function of the second kind
    Defining formula
    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
    Defining formula
    ℜ ⁡ absent {\displaystyle{\displaystyle\Re}}
    xml-id
    C1.S9.E2.m1afdec
    0 references
     
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