DLMF:13.23.E1 (Q4635): Difference between revisions

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Property / Symbols used
 
Property / Symbols used: Q11581 / rank
 
Normal rank
Property / Symbols used: Q11581 / qualifier
 
test:

F 1 2 ⁡ ( a , b ; c ; z ) Gauss-hypergeometric-F-as-2F1 𝑎 𝑏 𝑐 𝑧 {\displaystyle{\displaystyle{{}_{2}F_{1}}\left(\NVar{a},\NVar{b};\NVar{c};% \NVar{z}\right)}}

\genhyperF{2}{1}@{\NVar{a},\NVar{b}}{\NVar{c}}{\NVar{z}}
Property / Symbols used: Q11581 / qualifier
 
xml-id: C16.S2.m5adec
Property / Symbols used
 
Property / Symbols used: Whittaker confluent hypergeometric function / rank
 
Normal rank
Property / Symbols used: Whittaker confluent hypergeometric function / qualifier
 
test:

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

\WhittakerconfhyperM{\NVar{\kappa}}{\NVar{\mu}}@{\NVar{z}}
Property / Symbols used: Whittaker confluent hypergeometric function / qualifier
 
xml-id: C13.S14.E2.m2adec
Property / Symbols used
 
Property / Symbols used: Q10770 / rank
 
Normal rank
Property / Symbols used: Q10770 / qualifier
 
test:

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

\diff{\NVar{x}}
Property / Symbols used: Q10770 / qualifier
 
xml-id: C1.S4.SS4.m1adec
Property / Symbols used
 
Property / Symbols used: base of natural logarithm / rank
 
Normal rank
Property / Symbols used: base of natural logarithm / qualifier
 
test:

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

\expe
Property / Symbols used: base of natural logarithm / qualifier
 
xml-id: C4.S2.E11.m2adec
Property / Symbols used
 
Property / Symbols used: Q10771 / rank
 
Normal rank
Property / Symbols used: Q10771 / qualifier
 
test:

∫ {\displaystyle{\displaystyle\int}}

\int
Property / Symbols used: Q10771 / qualifier
 
xml-id: C1.S4.SS4.m3adec
Property / Symbols used
 
Property / Symbols used: Q10811 / rank
 
Normal rank
Property / Symbols used: Q10811 / qualifier
 
test:

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

\realpart@@
Property / Symbols used: Q10811 / qualifier
 
xml-id: C1.S9.E2.m1adec
Property / Symbols used
 
Property / Symbols used: Q11557 / rank
 
Normal rank
Property / Symbols used: Q11557 / qualifier
 
test:

z 𝑧 {\displaystyle{\displaystyle z}}

z
Property / Symbols used: Q11557 / qualifier
 
xml-id: C13.S1.XMD6.m1dec

Latest revision as of 15:22, 2 January 2020

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English
DLMF:13.23.E1
No description defined

    Statements

    test
    ∫ 0 ∞ e - z ⁢ t ⁢ t ν - 1 ⁢ M κ , μ ⁡ ( t ) ⁢ d t = Γ ⁡ ( μ + ν + 1 2 ) ( z + 1 2 ) μ + ν + 1 2 ⁢ F 1 2 ⁡ ( 1 2 + μ - κ , 1 2 + μ + ν 1 + 2 ⁢ μ ; 1 z + 1 2 ) , superscript subscript 0 superscript 𝑒 𝑧 𝑡 superscript 𝑡 𝜈 1 Whittaker-confluent-hypergeometric-M 𝜅 𝜇 𝑡 𝑡 Euler-Gamma 𝜇 𝜈 1 2 superscript 𝑧 1 2 𝜇 𝜈 1 2 Gauss-hypergeometric-F-as-2F1 1 2 𝜇 𝜅 1 2 𝜇 𝜈 1 2 𝜇 1 𝑧 1 2 {\displaystyle{\displaystyle\int_{0}^{\infty}e^{-zt}t^{\nu-1}M_{\kappa,\mu}% \left(t\right)\mathrm{d}t=\frac{\Gamma\left(\mu+\nu+\tfrac{1}{2}\right)}{\left% (z+\frac{1}{2}\right)^{\mu+\nu+\frac{1}{2}}}\*{{}_{2}F_{1}}\left({\tfrac{1}{2}% +\mu-\kappa,\tfrac{1}{2}+\mu+\nu\atop 1+2\mu};\frac{1}{z+\frac{1}{2}}\right),}}
    0 references
    DLMF id
    DLMF:13.23.E1
    0 references
    constraint
    ℜ ⁡ ( μ + ν + 1 2 ) > 0 𝜇 𝜈 1 2 0 {\displaystyle{\displaystyle\Re(\mu+\nu+\tfrac{1}{2})>0}}
    0 references
    ℜ ⁡ z > 1 2 𝑧 1 2 {\displaystyle{\displaystyle\Re z>\tfrac{1}{2}}}
    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
    Q11581
    test
    F 1 2 ⁡ ( a , b ; c ; z ) Gauss-hypergeometric-F-as-2F1 𝑎 𝑏 𝑐 𝑧 {\displaystyle{\displaystyle{{}_{2}F_{1}}\left(\NVar{a},\NVar{b};\NVar{c};% \NVar{z}\right)}}
    xml-id
    C16.S2.m5adec
    0 references
    Whittaker confluent hypergeometric function
    test
    M κ , μ ⁡ ( z ) Whittaker-confluent-hypergeometric-M 𝜅 𝜇 𝑧 {\displaystyle{\displaystyle M_{\NVar{\kappa},\NVar{\mu}}\left(\NVar{z}\right)}}
    xml-id
    C13.S14.E2.m2adec
    0 references
    Q10770
    test
    d x 𝑥 {\displaystyle{\displaystyle\mathrm{d}\NVar{x}}}
    xml-id
    C1.S4.SS4.m1adec
    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.m3adec
    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.m1dec
    0 references
     
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