DLMF:13.16.E2 (Q4553): Difference between revisions

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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.m2aadec
Property / Symbols used
 
Property / Symbols used: Whittaker confluent hypergeometric function / rank
 
Normal rank
Property / Symbols used: Whittaker confluent hypergeometric function / qualifier
 
Defining formula:

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.m2aadec
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.m1aadec
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.m2aadec
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.m1aadec
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.m1adec
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.m3aadec

Latest revision as of 16:10, 2 January 2020

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DLMF:13.16.E2
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    Statements

    Defining formula
    M κ , μ ⁡ ( z ) = Γ ⁡ ( 1 + 2 ⁢ μ ) ⁢ z λ Γ ⁡ ( 1 + 2 ⁢ μ - 2 ⁢ λ ) ⁢ Γ ⁡ ( 2 ⁢ λ ) ⁢ ∫ 0 1 M κ - λ , μ - λ ⁡ ( z ⁢ t ) ⁢ e 1 2 ⁢ z ⁢ ( t - 1 ) ⁢ t μ - λ - 1 2 ⁢ ( 1 - t ) 2 ⁢ λ - 1 ⁢ d t , Whittaker-confluent-hypergeometric-M 𝜅 𝜇 𝑧 Euler-Gamma 1 2 𝜇 superscript 𝑧 𝜆 Euler-Gamma 1 2 𝜇 2 𝜆 Euler-Gamma 2 𝜆 superscript subscript 0 1 Whittaker-confluent-hypergeometric-M 𝜅 𝜆 𝜇 𝜆 𝑧 𝑡 superscript 𝑒 1 2 𝑧 𝑡 1 superscript 𝑡 𝜇 𝜆 1 2 superscript 1 𝑡 2 𝜆 1 𝑡 {\displaystyle{\displaystyle M_{\kappa,\mu}\left(z\right)=\frac{\Gamma\left(1+% 2\mu\right)z^{\lambda}}{\Gamma\left(1+2\mu-2\lambda\right)\Gamma\left(2\lambda% \right)}\*\int_{0}^{1}M_{\kappa-\lambda,\mu-\lambda}\left(zt\right)e^{\frac{1}% {2}z(t-1)}t^{\mu-\lambda-\frac{1}{2}}{(1-t)^{2\lambda-1}}\mathrm{d}t,}}
    0 references
    DLMF id
    DLMF:13.16.E2
    0 references
    constraint
    ℜ ⁡ μ + 1 2 > ℜ ⁡ λ > 0 𝜇 1 2 𝜆 0 {\displaystyle{\displaystyle\Re\mu+\tfrac{1}{2}>\Re\lambda>0}}
    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.m2aadec
    0 references
    Whittaker confluent hypergeometric function
    Defining formula
    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.m2aadec
    0 references
    Q10770
    Defining formula
    d x 𝑥 {\displaystyle{\displaystyle\mathrm{d}\NVar{x}}}
    xml-id
    C1.S4.SS4.m1aadec
    0 references
    base of natural logarithm
    Defining formula
    e {\displaystyle{\displaystyle\mathrm{e}}}
    xml-id
    C4.S2.E11.m2aadec
    0 references
    Q10811
    Defining formula
    ℜ ⁡ absent {\displaystyle{\displaystyle\Re}}
    xml-id
    C1.S9.E2.m1aadec
    0 references
    Q11557
    Defining formula
    z 𝑧 {\displaystyle{\displaystyle z}}
    xml-id
    C13.S1.XMD6.m1adec
    0 references
    Q10771
    Defining formula
    ∫ {\displaystyle{\displaystyle\int}}
    xml-id
    C1.S4.SS4.m3aadec
    0 references
     
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