DLMF:13.10.E11 (Q4470)

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DLMF:13.10.E11
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    Statements

    Defining formula
    ∫ 0 ∞ t λ - 1 ⁢ U ⁡ ( a , b , t ) ⁢ d t = Γ ⁡ ( λ ) ⁢ Γ ⁡ ( a - λ ) ⁢ Γ ⁡ ( λ - b + 1 ) Γ ⁡ ( a ) ⁢ Γ ⁡ ( a - b + 1 ) , superscript subscript 0 superscript 𝑡 𝜆 1 Kummer-confluent-hypergeometric-U 𝑎 𝑏 𝑡 𝑡 Euler-Gamma 𝜆 Euler-Gamma 𝑎 𝜆 Euler-Gamma 𝜆 𝑏 1 Euler-Gamma 𝑎 Euler-Gamma 𝑎 𝑏 1 {\displaystyle{\displaystyle\int_{0}^{\infty}t^{\lambda-1}U\left(a,b,t\right)% \mathrm{d}t=\frac{\Gamma\left(\lambda\right)\Gamma\left(a-\lambda\right)\Gamma% \left(\lambda-b+1\right)}{\Gamma\left(a\right)\Gamma\left(a-b+1\right)},}}
    0 references
    DLMF id
    DLMF:13.10.E11
    0 references
    constraint
    max ⁡ ( ℜ ⁡ b - 1 , 0 ) < ℜ ⁡ λ < ℜ ⁡ a 𝑏 1 0 𝜆 𝑎 {\displaystyle{\displaystyle\max\left(\Re b-1,0\right)<\Re\lambda<\Re a}}
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    Symbols used
    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.m2addec
    0 references
    Q10770
    Defining formula
    d x 𝑥 {\displaystyle{\displaystyle\mathrm{d}\NVar{x}}}
    xml-id
    C1.S4.SS4.m1ajdec
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    Q10771
    Defining formula
    ∫ {\displaystyle{\displaystyle\int}}
    xml-id
    C1.S4.SS4.m3ajdec
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    Q10811
    Defining formula
    ℜ ⁡ absent {\displaystyle{\displaystyle\Re}}
    xml-id
    C1.S9.E2.m1ahdec
    0 references
    gamma function
    Defining formula
    Γ ⁡ ( z ) Euler-Gamma 𝑧 {\displaystyle{\displaystyle\Gamma\left(\NVar{z}\right)}}
    xml-id
    C5.S2.E1.m2agdec
    0 references
     
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