DLMF:13.4.E6 (Q4369)

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DLMF:13.4.E6
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    Statements

    test
    U ⁡ ( a , b , z ) = ( - 1 ) n ⁢ z 1 - b - n Γ ⁡ ( 1 + a - b ) ⁢ ∫ 0 ∞ 𝐌 ⁡ ( b - a , b , t ) ⁢ e - t ⁢ t b + n - 1 t + z ⁢ d t , Kummer-confluent-hypergeometric-U 𝑎 𝑏 𝑧 superscript 1 𝑛 superscript 𝑧 1 𝑏 𝑛 Euler-Gamma 1 𝑎 𝑏 superscript subscript 0 Kummer-confluent-hypergeometric-bold-M 𝑏 𝑎 𝑏 𝑡 superscript 𝑒 𝑡 superscript 𝑡 𝑏 𝑛 1 𝑡 𝑧 𝑡 {\displaystyle{\displaystyle U\left(a,b,z\right)=\frac{(-1)^{n}z^{1-b-n}}{% \Gamma\left(1+a-b\right)}\int_{0}^{\infty}\frac{{\mathbf{M}}\left(b-a,b,t% \right)e^{-t}t^{b+n-1}}{t+z}\mathrm{d}t,}}
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    DLMF id
    DLMF:13.4.E6
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    constraint
    | ph ⁡ z | < π phase 𝑧 𝜋 {\displaystyle{\displaystyle\left|\operatorname{ph}z\right|<\pi}}
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    - ℜ ⁡ b < n < 1 + ℜ ⁡ ( a - b ) 𝑏 𝑛 1 𝑎 𝑏 {\displaystyle{\displaystyle-\Re b<n<1+\Re(a-b)}}
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    | ph ⁡ z | < π ph 𝑧 𝜋 {\displaystyle{\displaystyle\left|\operatorname{ph}z\right|<\pi}}
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    n = 0 , 1 , 2 , … 𝑛 0 1 2 … {\displaystyle{\displaystyle n=0,1,2,\dots}}
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