DLMF:14.19.E7 (Q4920): Difference between revisions

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Property / Symbols used
 
Property / Symbols used: Q11481 / rank
 
Normal rank
Property / Symbols used: Q11481 / qualifier
 
Defining formula:

P ν μ ⁡ ( z ) Legendre-P-first-kind 𝜇 𝜈 𝑧 {\displaystyle{\displaystyle P^{\NVar{\mu}}_{\NVar{\nu}}\left(\NVar{z}\right)}}

\assLegendreP[\NVar{\mu}]{\NVar{\nu}}@{\NVar{z}}
Property / Symbols used: Q11481 / qualifier
 
xml-id: C14.S21.SS1.p1.m1abdec

Revision as of 13:30, 2 January 2020

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DLMF:14.19.E7
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    Statements

    Defining formula
    P n - 1 2 m ⁡ ( cosh ⁡ ξ ) = Γ ⁡ ( n + m + 1 2 ) Γ ⁡ ( n - m + 1 2 ) ⁢ ( 2 π ⁢ sinh ⁡ ξ ) 1 / 2 ⁢ 𝑸 m - 1 2 n ⁡ ( coth ⁡ ξ ) , Legendre-P-first-kind 𝑚 𝑛 1 2 𝜉 Euler-Gamma 𝑛 𝑚 1 2 Euler-Gamma 𝑛 𝑚 1 2 superscript 2 𝜋 𝜉 1 2 associated-Legendre-black-Q 𝑛 𝑚 1 2 hyperbolic-cotangent 𝜉 {\displaystyle{\displaystyle P^{m}_{n-\frac{1}{2}}\left(\cosh\xi\right)=\frac{% \Gamma\left(n+m+\tfrac{1}{2}\right)}{\Gamma\left(n-m+\tfrac{1}{2}\right)}\*% \left(\frac{2}{\pi\sinh\xi}\right)^{1/2}\boldsymbol{Q}^{n}_{m-\frac{1}{2}}% \left(\coth\xi\right),}}
    0 references
    DLMF id
    DLMF:14.19.E7
    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.m2addec
    0 references
    Q11481
    Defining formula
    P ν μ ⁡ ( z ) Legendre-P-first-kind 𝜇 𝜈 𝑧 {\displaystyle{\displaystyle P^{\NVar{\mu}}_{\NVar{\nu}}\left(\NVar{z}\right)}}
    xml-id
    C14.S21.SS1.p1.m1abdec
    0 references
     
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