DLMF:13.23.E3 (Q4637): Difference between revisions

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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.m3abdec
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.m1abdec
Property / Symbols used
 
Property / Symbols used: gamma function / rank
 
Normal rank
Property / Symbols used: gamma function / qualifier
 
test:

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.m2abdec
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.m2abdec
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.m1abdec

Latest revision as of 15:22, 2 January 2020

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Language Label Description Also known as
English
DLMF:13.23.E3
No description defined

    Statements

    test
    1 Γ ⁡ ( 1 + 2 ⁢ μ ) ⁢ ∫ 0 ∞ e - 1 2 ⁢ t ⁢ t ν - 1 ⁢ M κ , μ ⁡ ( t ) ⁢ d t = Γ ⁡ ( μ + ν + 1 2 ) ⁢ Γ ⁡ ( κ - ν ) Γ ⁡ ( 1 2 + μ + κ ) ⁢ Γ ⁡ ( 1 2 + μ - ν ) , 1 Euler-Gamma 1 2 𝜇 superscript subscript 0 superscript 𝑒 1 2 𝑡 superscript 𝑡 𝜈 1 Whittaker-confluent-hypergeometric-M 𝜅 𝜇 𝑡 𝑡 Euler-Gamma 𝜇 𝜈 1 2 Euler-Gamma 𝜅 𝜈 Euler-Gamma 1 2 𝜇 𝜅 Euler-Gamma 1 2 𝜇 𝜈 {\displaystyle{\displaystyle\frac{1}{\Gamma\left(1+2\mu\right)}\int_{0}^{% \infty}e^{-\frac{1}{2}t}t^{\nu-1}M_{\kappa,\mu}\left(t\right)\mathrm{d}t=\frac% {\Gamma\left(\mu+\nu+\frac{1}{2}\right)\Gamma\left(\kappa-\nu\right)}{\Gamma% \left(\frac{1}{2}+\mu+\kappa\right)\Gamma\left(\frac{1}{2}+\mu-\nu\right)},}}
    0 references
    DLMF id
    DLMF:13.23.E3
    0 references
    constraint
    - 1 2 - ℜ ⁡ μ < ℜ ⁡ ν < ℜ ⁡ κ 1 2 𝜇 𝜈 𝜅 {\displaystyle{\displaystyle-\tfrac{1}{2}-\Re\mu<\Re\nu<\Re\kappa}}
    0 references
    Symbols used
    base of natural logarithm
    test
    e {\displaystyle{\displaystyle\mathrm{e}}}
    xml-id
    C4.S2.E11.m2abdec
    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.m1abdec
    0 references
    gamma function
    test
    Γ ⁡ ( z ) Euler-Gamma 𝑧 {\displaystyle{\displaystyle\Gamma\left(\NVar{z}\right)}}
    xml-id
    C5.S2.E1.m2abdec
    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.m2abdec
    0 references
    Q10770
    test
    d x 𝑥 {\displaystyle{\displaystyle\mathrm{d}\NVar{x}}}
    xml-id
    C1.S4.SS4.m1abdec
    0 references
     
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