DLMF:13.23.E4 (Q4638): Difference between revisions

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Property / constraint
 

ℜ ⁡ z > - 1 2 𝑧 1 2 {\displaystyle{\displaystyle\Re z>-\tfrac{1}{2}}}

\realpart@@{z}>-\tfrac{1}{2}
Property / constraint: ℜ ⁡ z > - 1 2 𝑧 1 2 {\displaystyle{\displaystyle\Re z>-\tfrac{1}{2}}} / rank
 
Normal rank
Property / Symbols used
 
Property / Symbols used: gamma function / rank
 
Normal rank
Property / Symbols used: gamma function / qualifier
 
Defining formula:

Γ ⁡ ( z ) Euler-Gamma 𝑧 {\displaystyle{\displaystyle\Gamma\left(\NVar{z}\right)}}

\EulerGamma@{\NVar{z}}
Property / Symbols used: gamma function / qualifier
 
xml-id: C5.S2.E1.m2acdec
Property / Symbols used
 
Property / Symbols used: Whittaker confluent hypergeometric function / rank
 
Normal rank
Property / Symbols used: Whittaker confluent hypergeometric function / qualifier
 
Defining formula:

W κ , μ ⁡ ( z ) Whittaker-confluent-hypergeometric-W 𝜅 𝜇 𝑧 {\displaystyle{\displaystyle W_{\NVar{\kappa},\NVar{\mu}}\left(\NVar{z}\right)}}

\WhittakerconfhyperW{\NVar{\kappa}}{\NVar{\mu}}@{\NVar{z}}
Property / Symbols used: Whittaker confluent hypergeometric function / qualifier
 
xml-id: C13.S14.E3.m2adec
Property / Symbols used
 
Property / Symbols used: Q10770 / rank
 
Normal rank
Property / Symbols used: Q10770 / qualifier
 
Defining formula:

d x 𝑥 {\displaystyle{\displaystyle\mathrm{d}\NVar{x}}}

\diff{\NVar{x}}
Property / Symbols used: Q10770 / qualifier
 
xml-id: C1.S4.SS4.m1acdec
Property / Symbols used
 
Property / Symbols used: base of natural logarithm / rank
 
Normal rank
Property / Symbols used: base of natural logarithm / qualifier
 
Defining formula:

e {\displaystyle{\displaystyle\mathrm{e}}}

\expe
Property / Symbols used: base of natural logarithm / qualifier
 
xml-id: C4.S2.E11.m2acdec
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
 
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}}}

\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
Property / Symbols used
 
Property / Symbols used: Q10771 / rank
 
Normal rank
Property / Symbols used: Q10771 / qualifier
 
Defining formula:

∫ {\displaystyle{\displaystyle\int}}

\int
Property / Symbols used: Q10771 / qualifier
 
xml-id: C1.S4.SS4.m3acdec
Property / Symbols used
 
Property / Symbols used: Q10811 / rank
 
Normal rank
Property / Symbols used: Q10811 / qualifier
 
Defining formula:

ℜ ⁡ absent {\displaystyle{\displaystyle\Re}}

\realpart@@
Property / Symbols used: Q10811 / qualifier
 
xml-id: C1.S9.E2.m1acdec
Property / Symbols used
 
Property / Symbols used: Q11557 / rank
 
Normal rank
Property / Symbols used: Q11557 / qualifier
 
Defining formula:

z 𝑧 {\displaystyle{\displaystyle z}}

z
Property / Symbols used: Q11557 / qualifier
 
xml-id: C13.S1.XMD6.m1bdec

Latest revision as of 01:02, 2 January 2020

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DLMF:13.23.E4
No description defined

    Statements

    Defining formula
    ∫ 0 ∞ e - z ⁢ t ⁢ t ν - 1 ⁢ W κ , μ ⁡ ( t ) ⁢ d t = Γ ⁡ ( 1 2 + μ + ν ) ⁢ Γ ⁡ ( 1 2 - μ + ν ) ⁢ 𝐅 1 2 ⁡ ( 1 2 - μ + ν , 1 2 + μ + ν ν - κ + 1 ; 1 2 - z ) , superscript subscript 0 superscript 𝑒 𝑧 𝑡 superscript 𝑡 𝜈 1 Whittaker-confluent-hypergeometric-W 𝜅 𝜇 𝑡 𝑡 Euler-Gamma 1 2 𝜇 𝜈 Euler-Gamma 1 2 𝜇 𝜈 hypergeometric-bold-pFq 2 1 1 2 𝜇 𝜈 1 2 𝜇 𝜈 𝜈 𝜅 1 1 2 𝑧 {\displaystyle{\displaystyle\int_{0}^{\infty}e^{-zt}t^{\nu-1}W_{\kappa,\mu}% \left(t\right)\mathrm{d}t=\Gamma\left(\tfrac{1}{2}+\mu+\nu\right)\Gamma\left(% \tfrac{1}{2}-\mu+\nu\right)\*{{}_{2}{\mathbf{F}}_{1}}\left({\tfrac{1}{2}-\mu+% \nu,\tfrac{1}{2}+\mu+\nu\atop\nu-\kappa+1};\tfrac{1}{2}-z\right),}}
    0 references
    DLMF id
    DLMF:13.23.E4
    0 references
    constraint
    ℜ ⁡ ( ν + 1 2 ) > | ℜ ⁡ μ | 𝜈 1 2 𝜇 {\displaystyle{\displaystyle\Re(\nu+\tfrac{1}{2})>|\Re\mu|}}
    0 references
    ℜ ⁡ z > - 1 2 𝑧 1 2 {\displaystyle{\displaystyle\Re z>-\tfrac{1}{2}}}
    0 references
    Symbols used
    gamma function
    Defining formula
    Γ ⁡ ( z ) Euler-Gamma 𝑧 {\displaystyle{\displaystyle\Gamma\left(\NVar{z}\right)}}
    xml-id
    C5.S2.E1.m2acdec
    0 references
    Whittaker confluent hypergeometric function
    Defining formula
    W κ , μ ⁡ ( z ) Whittaker-confluent-hypergeometric-W 𝜅 𝜇 𝑧 {\displaystyle{\displaystyle W_{\NVar{\kappa},\NVar{\mu}}\left(\NVar{z}\right)}}
    xml-id
    C13.S14.E3.m2adec
    0 references
    Q10770
    Defining formula
    d x 𝑥 {\displaystyle{\displaystyle\mathrm{d}\NVar{x}}}
    xml-id
    C1.S4.SS4.m1acdec
    0 references
    base of natural logarithm
    Defining formula
    e {\displaystyle{\displaystyle\mathrm{e}}}
    xml-id
    C4.S2.E11.m2acdec
    0 references
    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.m2adec
    0 references
    Q10771
    Defining formula
    ∫ {\displaystyle{\displaystyle\int}}
    xml-id
    C1.S4.SS4.m3acdec
    0 references
    Q10811
    Defining formula
    ℜ ⁡ absent {\displaystyle{\displaystyle\Re}}
    xml-id
    C1.S9.E2.m1acdec
    0 references
    Q11557
    Defining formula
    z 𝑧 {\displaystyle{\displaystyle z}}
    xml-id
    C13.S1.XMD6.m1bdec
    0 references
     
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