DLMF:13.4.E9 (Q4372)

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DLMF:13.4.E9
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    Statements

    Defining formula
    𝐌 ⁑ ( a , b , z ) = Ξ“ ⁑ ( 1 + a - b ) 2 ⁒ Ο€ ⁒ i ⁒ Ξ“ ⁑ ( a ) ⁒ ∫ 0 ( 1 + ) e z ⁒ t ⁒ t a - 1 ⁒ ( t - 1 ) b - a - 1 ⁒ d t , Kummer-confluent-hypergeometric-bold-M π‘Ž 𝑏 𝑧 Euler-Gamma 1 π‘Ž 𝑏 2 πœ‹ imaginary-unit Euler-Gamma π‘Ž superscript subscript 0 limit-from 1 superscript 𝑒 𝑧 𝑑 superscript 𝑑 π‘Ž 1 superscript 𝑑 1 𝑏 π‘Ž 1 𝑑 {\displaystyle{\displaystyle{\mathbf{M}}\left(a,b,z\right)=\frac{\Gamma\left(1% +a-b\right)}{2\pi\mathrm{i}\Gamma\left(a\right)}\int_{0}^{(1+)}e^{zt}t^{a-1}{(% t-1)^{b-a-1}}\mathrm{d}t,}}
    0 references
    DLMF id
    DLMF:13.4.E9
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    constraint
    β„œ ⁑ a > 0 π‘Ž 0 {\displaystyle{\displaystyle\Re a>0}}
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    b - a β‰  1 , 2 , 3 , … 𝑏 π‘Ž 1 2 3 … {\displaystyle{\displaystyle b-a\neq 1,2,3,\dots}}
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    β„œ ⁑ a > 0 π‘Ž 0 {\displaystyle{\displaystyle\Re a>0}}
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    Symbols used
    gamma function
    Defining formula
    Ξ“ ⁑ ( z ) Euler-Gamma 𝑧 {\displaystyle{\displaystyle\Gamma\left(\NVar{z}\right)}}
    xml-id
    C5.S2.E1.m2agdec
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    Olver’s confluent hypergeometric function
    Defining formula
    𝐌 ⁑ ( a , b , z ) Kummer-confluent-hypergeometric-bold-M π‘Ž 𝑏 𝑧 {\displaystyle{\displaystyle{\mathbf{M}}\left(\NVar{a},\NVar{b},\NVar{z}\right% )}}
    xml-id
    C13.S2.E3.m2addec
    0 references
    the ratio of the circumference of a circle to its diameter
    Defining formula
    Ο€ {\displaystyle{\displaystyle\pi}}
    xml-id
    C3.S12.E1.m2addec
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    Q10770
    Defining formula
    d x π‘₯ {\displaystyle{\displaystyle\mathrm{d}\NVar{x}}}
    xml-id
    C1.S4.SS4.m1ahdec
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    base of natural logarithm
    Defining formula
    e {\displaystyle{\displaystyle\mathrm{e}}}
    xml-id
    C4.S2.E11.m2agdec
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    imaginary unit
    Defining formula
    i imaginary-unit {\displaystyle{\displaystyle\mathrm{i}}}
    xml-id
    C1.S9.E1.m2adec
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    Q10771
    Defining formula
    ∫ {\displaystyle{\displaystyle\int}}
    xml-id
    C1.S4.SS4.m3ahdec
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