DLMF:14.9.E15 (Q4807): Difference between revisions

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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.m3abdec

Revision as of 01:38, 2 January 2020

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DLMF:14.9.E15
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    Statements

    Defining formula
    2 ⁢ sin ⁡ ( μ ⁢ π ) π ⁢ 𝑸 ν μ ⁡ ( x ) = P ν μ ⁡ ( x ) Γ ⁡ ( ν + μ + 1 ) - P ν - μ ⁡ ( x ) Γ ⁡ ( ν - μ + 1 ) . 2 𝜇 𝜋 𝜋 associated-Legendre-black-Q 𝜇 𝜈 𝑥 Legendre-P-first-kind 𝜇 𝜈 𝑥 Euler-Gamma 𝜈 𝜇 1 Legendre-P-first-kind 𝜇 𝜈 𝑥 Euler-Gamma 𝜈 𝜇 1 {\displaystyle{\displaystyle\frac{2\sin\left(\mu\pi\right)}{\pi}\boldsymbol{Q}% ^{\mu}_{\nu}\left(x\right)=\frac{P^{\mu}_{\nu}\left(x\right)}{\Gamma\left(\nu+% \mu+1\right)}-\frac{P^{-\mu}_{\nu}\left(x\right)}{\Gamma\left(\nu-\mu+1\right)% }.}}
    0 references
    DLMF id
    DLMF:14.9.E15
    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.m2ahdec
    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.m1acdec
    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.m3abdec
    0 references
     
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