DLMF:14.19.E5 (Q4918): Difference between revisions

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‎Changed an Item: Add constraint
‎Changed an Item: Add constraint
Property / Symbols used
 
Property / Symbols used: Q11592 / rank
 
Normal rank
Property / Symbols used: Q11592 / qualifier
 
Defining formula:

𝑸 ν μ ⁡ ( z ) associated-Legendre-black-Q 𝜇 𝜈 𝑧 {\displaystyle{\displaystyle\boldsymbol{Q}^{\NVar{\mu}}_{\NVar{\nu}}\left(% \NVar{z}\right)}}

\assLegendreOlverQ[\NVar{\mu}]{\NVar{\nu}}@{\NVar{z}}
Property / Symbols used: Q11592 / qualifier
 
xml-id: C14.S21.SS1.p1.m3aadec

Revision as of 13:30, 2 January 2020

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DLMF:14.19.E5
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    Statements

    Defining formula
    𝑸 n - 1 2 m ⁡ ( cosh ⁡ ξ ) = Γ ⁡ ( n + 1 2 ) Γ ⁡ ( n + m + 1 2 ) ⁢ Γ ⁡ ( n - m + 1 2 ) ⁢ ∫ 0 ∞ cosh ⁡ ( m ⁢ t ) ( cosh ⁡ ξ + cosh ⁡ t ⁢ sinh ⁡ ξ ) n + ( 1 / 2 ) ⁢ d t , associated-Legendre-black-Q 𝑚 𝑛 1 2 𝜉 Euler-Gamma 𝑛 1 2 Euler-Gamma 𝑛 𝑚 1 2 Euler-Gamma 𝑛 𝑚 1 2 superscript subscript 0 𝑚 𝑡 superscript 𝜉 𝑡 𝜉 𝑛 1 2 𝑡 {\displaystyle{\displaystyle\boldsymbol{Q}^{m}_{n-\frac{1}{2}}\left(\cosh\xi% \right)=\frac{\Gamma\left(n+\frac{1}{2}\right)}{\Gamma\left(n+m+\tfrac{1}{2}% \right)\Gamma\left(n-m+\frac{1}{2}\right)}\*\int_{0}^{\infty}\frac{\cosh\left(% mt\right)}{(\cosh\xi+\cosh t\sinh\xi)^{n+(1/2)}}\mathrm{d}t,}}
    0 references
    DLMF id
    DLMF:14.19.E5
    0 references
    constraint
    m < n + 1 2 𝑚 𝑛 1 2 {\displaystyle{\displaystyle m<n+\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.m2abdec
    0 references
    Q11592
    Defining formula
    𝑸 ν μ ⁡ ( z ) associated-Legendre-black-Q 𝜇 𝜈 𝑧 {\displaystyle{\displaystyle\boldsymbol{Q}^{\NVar{\mu}}_{\NVar{\nu}}\left(% \NVar{z}\right)}}
    xml-id
    C14.S21.SS1.p1.m3aadec
    0 references
     
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