DLMF:13.10.E3 (Q4462): Difference between revisions

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Property / Symbols used
 
Property / Symbols used: scaled (or Olver’s) generalized hypergeometric function / rank
Normal rank
 
Property / Symbols used: scaled (or Olver’s) generalized hypergeometric function / qualifier
test:

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}}}

\genhyperOlverF{\NVar{p}}{\NVar{q}}@{\NVar{\mathbf{a}}}{\NVar{\mathbf{b}}}{% \NVar{z}}
 
Property / Symbols used: scaled (or Olver’s) generalized hypergeometric function / qualifier
xml-id: C16.S2.E5.m2adec
 

Revision as of 15:00, 2 January 2020

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DLMF:13.10.E3
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    Statements

    test
    ∫ 0 ∞ e - z ⁢ t ⁢ t b - 1 ⁢ 𝐌 ⁡ ( a , c , k ⁢ t ) ⁢ d t = Γ ⁡ ( b ) ⁢ z - b ⁢ 𝐅 1 2 ⁡ ( a , b ; c ; k / z ) , superscript subscript 0 superscript 𝑒 𝑧 𝑡 superscript 𝑡 𝑏 1 Kummer-confluent-hypergeometric-bold-M 𝑎 𝑐 𝑘 𝑡 𝑡 Euler-Gamma 𝑏 superscript 𝑧 𝑏 hypergeometric-bold-pFq 2 1 𝑎 𝑏 𝑐 𝑘 𝑧 {\displaystyle{\displaystyle\int_{0}^{\infty}e^{-zt}t^{b-1}{\mathbf{M}}\left(a% ,c,kt\right)\mathrm{d}t=\Gamma\left(b\right)z^{-b}{{}_{2}{\mathbf{F}}_{1}}% \left(a,b;c;\ifrac{k}{z}\right),}}
    0 references
    DLMF id
    DLMF:13.10.E3
    0 references
    constraint
    ℜ ⁡ b > 0 𝑏 0 {\displaystyle{\displaystyle\Re b>0}}
    0 references
    ℜ ⁡ z > max ⁡ ( ℜ ⁡ k , 0 ) 𝑧 𝑘 0 {\displaystyle{\displaystyle\Re z>\max\left(\Re k,0\right)}}
    0 references
    Symbols used
    gamma function
    test
    Γ ⁡ ( z ) Euler-Gamma 𝑧 {\displaystyle{\displaystyle\Gamma\left(\NVar{z}\right)}}
    xml-id
    C5.S2.E1.m2adec
    0 references
    Olver’s confluent hypergeometric function
    test
    𝐌 ⁡ ( a , b , z ) Kummer-confluent-hypergeometric-bold-M 𝑎 𝑏 𝑧 {\displaystyle{\displaystyle{\mathbf{M}}\left(\NVar{a},\NVar{b},\NVar{z}\right% )}}
    xml-id
    C13.S2.E3.m2aadec
    0 references
    Q10770
    test
    d x 𝑥 {\displaystyle{\displaystyle\mathrm{d}\NVar{x}}}
    xml-id
    C1.S4.SS4.m1abdec
    0 references
    base of natural logarithm
    test
    e {\displaystyle{\displaystyle\mathrm{e}}}
    xml-id
    C4.S2.E11.m2adec
    0 references
    Q10771
    test
    ∫ {\displaystyle{\displaystyle\int}}
    xml-id
    C1.S4.SS4.m3abdec
    0 references
    Q10811
    test
    ℜ ⁡ absent {\displaystyle{\displaystyle\Re}}
    xml-id
    C1.S9.E2.m1adec
    0 references
    Q11557
    test
    z 𝑧 {\displaystyle{\displaystyle z}}
    xml-id
    C13.S1.XMD6.m1bdec
    0 references
    scaled (or Olver’s) generalized hypergeometric function
    test
    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.m2adec
    0 references
     
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