DLMF:13.4.E15 (Q4378)

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DLMF:13.4.E15
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    Statements

    Defining formula
    U ⁡ ( a , b , z ) Γ ⁡ ( c ) ⁢ Γ ⁡ ( c - b + 1 ) = z 1 - c 2 ⁢ π ⁢ i ⁢ ∫ - ∞ ( 0 + ) e z ⁢ t ⁢ t - c ⁢ 𝐅 1 2 ⁡ ( a , c ; a + c - b + 1 ; 1 - 1 t ) ⁢ d t , Kummer-confluent-hypergeometric-U 𝑎 𝑏 𝑧 Euler-Gamma 𝑐 Euler-Gamma 𝑐 𝑏 1 superscript 𝑧 1 𝑐 2 𝜋 imaginary-unit superscript subscript limit-from 0 superscript 𝑒 𝑧 𝑡 superscript 𝑡 𝑐 hypergeometric-bold-pFq 2 1 𝑎 𝑐 𝑎 𝑐 𝑏 1 1 1 𝑡 𝑡 {\displaystyle{\displaystyle\frac{U\left(a,b,z\right)}{\Gamma\left(c\right)% \Gamma\left(c-b+1\right)}=\frac{z^{1-c}}{2\pi\mathrm{i}}\int_{-\infty}^{(0+)}e% ^{zt}t^{-c}{{}_{2}{\mathbf{F}}_{1}}\left(a,c;a+c-b+1;1-\frac{1}{t}\right)% \mathrm{d}t,}}
    0 references
    DLMF id
    DLMF:13.4.E15
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    constraint
    | ph ⁡ z | < 1 2 ⁢ π phase 𝑧 1 2 𝜋 {\displaystyle{\displaystyle\left|\operatorname{ph}z\right|<\frac{1}{2}\pi}}
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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.m2aldec
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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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    the ratio of the circumference of a circle to its diameter
    Defining formula
    π {\displaystyle{\displaystyle\pi}}
    xml-id
    C3.S12.E1.m2ajdec
    0 references
    Q10770
    Defining formula
    d x 𝑥 {\displaystyle{\displaystyle\mathrm{d}\NVar{x}}}
    xml-id
    C1.S4.SS4.m1andec
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    base of natural logarithm
    Defining formula
    e {\displaystyle{\displaystyle\mathrm{e}}}
    xml-id
    C4.S2.E11.m2amdec
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    scaled (or Olver’s) generalized hypergeometric function
    Defining formula
    𝐅 q p ⁡ ( 𝐚 ; 𝐛 ; ) hypergeometric-bold-pFq 𝑝 𝑞 𝐚 𝐛 {\displaystyle{\displaystyle{{}_{\NVar{p}}{\mathbf{F}}_{\NVar{q}}}\left(\NVar{% \mathbf{a}};\NVar{\mathbf{b}};\right)\)\@add@PDF@RDFa@triples\end{document}}}
    xml-id
    C16.S2.E5.m2abdec
    0 references
    imaginary unit
    Defining formula
    i imaginary-unit {\displaystyle{\displaystyle\mathrm{i}}}
    xml-id
    C1.S9.E1.m2afdec
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    Q10771
    Defining formula
    ∫ {\displaystyle{\displaystyle\int}}
    xml-id
    C1.S4.SS4.m3andec
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    Q10817
    Defining formula
    ph phase {\displaystyle{\displaystyle\operatorname{ph}}}
    xml-id
    C1.S9.E7.m1agdec
    0 references
    Q11557
    Defining formula
    z 𝑧 {\displaystyle{\displaystyle z}}
    xml-id
    C13.S1.XMD6.m1ndec
    0 references
     
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