DLMF:13.4.E12 (Q4375)

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DLMF:13.4.E12
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    Statements

    Defining formula
    𝐌 ⁑ ( a , c , z ) = Ξ“ ⁑ ( b ) 2 ⁒ Ο€ ⁒ i ⁒ z 1 - b ⁒ ∫ - ∞ ( 0 + , 1 + ) e z ⁒ t ⁒ t - b ⁒ 𝐅 1 2 ⁑ ( a , b ; c ; 1 / t ) ⁒ d t , Kummer-confluent-hypergeometric-bold-M π‘Ž 𝑐 𝑧 Euler-Gamma 𝑏 2 πœ‹ imaginary-unit superscript 𝑧 1 𝑏 superscript subscript limit-from 0 limit-from 1 superscript 𝑒 𝑧 𝑑 superscript 𝑑 𝑏 hypergeometric-bold-pFq 2 1 π‘Ž 𝑏 𝑐 1 𝑑 𝑑 {\displaystyle{\displaystyle{\mathbf{M}}\left(a,c,z\right)=\frac{\Gamma\left(b% \right)}{2\pi\mathrm{i}}z^{1-b}\int_{-\infty}^{(0+,1+)}e^{zt}t^{-b}{{}_{2}{% \mathbf{F}}_{1}}\left(a,b;c;\ifrac{1}{t}\right)\mathrm{d}t,}}
    0 references
    DLMF id
    DLMF:13.4.E12
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    constraint
    | ph ⁑ z | < 1 2 ⁒ Ο€ phase 𝑧 1 2 πœ‹ {\displaystyle{\displaystyle\left|\operatorname{ph}z\right|<\frac{1}{2}\pi}}
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    b β‰  0 , - 1 , - 2 , … 𝑏 0 1 2 … {\displaystyle{\displaystyle b\neq 0,-1,-2,\dots}}
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    | ph ⁑ z | < 1 2 ⁒ Ο€ ph 𝑧 1 2 πœ‹ {\displaystyle{\displaystyle\left|\operatorname{ph}z\right|<\frac{1}{2}\pi}}
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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.m2ajdec
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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.m2agdec
    0 references
    the ratio of the circumference of a circle to its diameter
    Defining formula
    Ο€ {\displaystyle{\displaystyle\pi}}
    xml-id
    C3.S12.E1.m2agdec
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    Q10770
    Defining formula
    d x π‘₯ {\displaystyle{\displaystyle\mathrm{d}\NVar{x}}}
    xml-id
    C1.S4.SS4.m1akdec
    0 references
    base of natural logarithm
    Defining formula
    e {\displaystyle{\displaystyle\mathrm{e}}}
    xml-id
    C4.S2.E11.m2ajdec
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    scaled (or Olver’s) generalized hypergeometric function
    Defining formula
    𝐅 q p ⁑ ( 𝐚 ; 𝐛 ; ) hypergeometric-bold-pFq 𝑝 π‘ž 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.m2aadec
    0 references
    imaginary unit
    Defining formula
    i imaginary-unit {\displaystyle{\displaystyle\mathrm{i}}}
    xml-id
    C1.S9.E1.m2acdec
    0 references
    Q10771
    Defining formula
    ∫ {\displaystyle{\displaystyle\int}}
    xml-id
    C1.S4.SS4.m3akdec
    0 references
     
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