DLMF:14.5.E4 (Q4716): Difference between revisions

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‎Changed an Item: Add constraint
‎Changed an Item: Add constraint
Property / Symbols used
 
Property / Symbols used: Ferrers function of the second kind / rank
 
Normal rank
Property / Symbols used: Ferrers function of the second kind / qualifier
 
Defining formula:

𝖰 ν μ ⁡ ( x ) Ferrers-Legendre-Q-first-kind 𝜇 𝜈 𝑥 {\displaystyle{\displaystyle\mathsf{Q}^{\NVar{\mu}}_{\NVar{\nu}}\left(\NVar{x}% \right)}}

\FerrersQ[\NVar{\mu}]{\NVar{\nu}}@{\NVar{x}}
Property / Symbols used: Ferrers function of the second kind / qualifier
 
xml-id: C14.S3.E2.m2aadec

Revision as of 01:21, 2 January 2020

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DLMF:14.5.E4
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    Statements

    Defining formula
    d 𝖰 ν μ ⁡ ( x ) d x | x = 0 = 2 μ ⁢ π 1 / 2 ⁢ cos ⁡ ( 1 2 ⁢ ( ν + μ ) ⁢ π ) ⁢ Γ ⁡ ( 1 2 ⁢ ν + 1 2 ⁢ μ + 1 ) Γ ⁡ ( 1 2 ⁢ ν - 1 2 ⁢ μ + 1 2 ) , evaluated-at derivative Ferrers-Legendre-Q-first-kind 𝜇 𝜈 𝑥 𝑥 𝑥 0 superscript 2 𝜇 superscript 𝜋 1 2 1 2 𝜈 𝜇 𝜋 Euler-Gamma 1 2 𝜈 1 2 𝜇 1 Euler-Gamma 1 2 𝜈 1 2 𝜇 1 2 {\displaystyle{\displaystyle\left.\frac{\mathrm{d}\mathsf{Q}^{\mu}_{\nu}\left(% x\right)}{\mathrm{d}x}\right|_{x=0}=\frac{2^{\mu}\pi^{1/2}\cos\left(\frac{1}{2% }(\nu+\mu)\pi\right)\Gamma\left(\frac{1}{2}\nu+\frac{1}{2}\mu+1\right)}{\Gamma% \left(\frac{1}{2}\nu-\frac{1}{2}\mu+\frac{1}{2}\right)},}}
    0 references
    DLMF id
    DLMF:14.5.E4
    0 references
    constraint
    ν + μ ≠ - 2 , - 4 , - 6 , … 𝜈 𝜇 2 4 6 … {\displaystyle{\displaystyle\nu+\mu\neq-2,-4,-6,\dots}}
    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.m2acdec
    0 references
    Ferrers function of the second kind
    Defining formula
    𝖰 ν μ ⁡ ( x ) Ferrers-Legendre-Q-first-kind 𝜇 𝜈 𝑥 {\displaystyle{\displaystyle\mathsf{Q}^{\NVar{\mu}}_{\NVar{\nu}}\left(\NVar{x}% \right)}}
    xml-id
    C14.S3.E2.m2aadec
    0 references
     
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