DLMF:13.4.E10 (Q4373)

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DLMF:13.4.E10
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    Statements

    Defining formula
    𝐌 ⁑ ( a , b , z ) = e - a ⁒ Ο€ ⁒ i ⁒ Ξ“ ⁑ ( 1 - a ) 2 ⁒ Ο€ ⁒ i ⁒ Ξ“ ⁑ ( b - a ) ⁒ ∫ 1 ( 0 + ) e z ⁒ t ⁒ t a - 1 ⁒ ( 1 - t ) b - a - 1 ⁒ d t , Kummer-confluent-hypergeometric-bold-M π‘Ž 𝑏 𝑧 superscript 𝑒 π‘Ž πœ‹ imaginary-unit Euler-Gamma 1 π‘Ž 2 πœ‹ imaginary-unit Euler-Gamma 𝑏 π‘Ž superscript subscript 1 limit-from 0 superscript 𝑒 𝑧 𝑑 superscript 𝑑 π‘Ž 1 superscript 1 𝑑 𝑏 π‘Ž 1 𝑑 {\displaystyle{\displaystyle{\mathbf{M}}\left(a,b,z\right)=e^{-a\pi\mathrm{i}}% \frac{\Gamma\left(1-a\right)}{2\pi\mathrm{i}\Gamma\left(b-a\right)}\int_{1}^{(% 0+)}e^{zt}t^{a-1}{(1-t)^{b-a-1}}\mathrm{d}t,}}
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    DLMF id
    DLMF:13.4.E10
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    constraint
    β„œ ⁑ ( b - a ) > 0 𝑏 π‘Ž 0 {\displaystyle{\displaystyle\Re(b-a)>0}}
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    a β‰  1 , 2 , 3 , … π‘Ž 1 2 3 … {\displaystyle{\displaystyle a\neq 1,2,3,\dots}}
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    β„œ ⁑ ( b - a ) > 0 𝑏 π‘Ž 0 {\displaystyle{\displaystyle\Re(b-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.m2ahdec
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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.m2aedec
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    the ratio of the circumference of a circle to its diameter
    Defining formula
    Ο€ {\displaystyle{\displaystyle\pi}}
    xml-id
    C3.S12.E1.m2aedec
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    Q10770
    Defining formula
    d x π‘₯ {\displaystyle{\displaystyle\mathrm{d}\NVar{x}}}
    xml-id
    C1.S4.SS4.m1aidec
    0 references
     
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