DLMF:14.3.E2 (Q4691): Difference between revisions

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Property / Symbols used
 
Property / Symbols used: Q11591 / rank
 
Normal rank
Property / Symbols used: Q11591 / qualifier
 
Defining formula:

μ 𝜇 {\displaystyle{\displaystyle\mu}}

\mu
Property / Symbols used: Q11591 / qualifier
 
xml-id: C14.S1.XMD7.m1adec

Revision as of 01:12, 2 January 2020

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DLMF:14.3.E2
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    Statements

    Defining formula
    𝖰 ν μ ⁡ ( x ) = π 2 ⁢ sin ⁡ ( μ ⁢ π ) ⁢ ( cos ⁡ ( μ ⁢ π ) ⁢ ( 1 + x 1 - x ) μ / 2 ⁢ 𝐅 ⁡ ( ν + 1 , - ν ; 1 - μ ; 1 2 - 1 2 ⁢ x ) - Γ ⁡ ( ν + μ + 1 ) Γ ⁡ ( ν - μ + 1 ) ⁢ ( 1 - x 1 + x ) μ / 2 ⁢ 𝐅 ⁡ ( ν + 1 , - ν ; 1 + μ ; 1 2 - 1 2 ⁢ x ) ) . Ferrers-Legendre-Q-first-kind 𝜇 𝜈 𝑥 𝜋 2 𝜇 𝜋 𝜇 𝜋 superscript 1 𝑥 1 𝑥 𝜇 2 scaled-hypergeometric-bold-F 𝜈 1 𝜈 1 𝜇 1 2 1 2 𝑥 Euler-Gamma 𝜈 𝜇 1 Euler-Gamma 𝜈 𝜇 1 superscript 1 𝑥 1 𝑥 𝜇 2 scaled-hypergeometric-bold-F 𝜈 1 𝜈 1 𝜇 1 2 1 2 𝑥 {\displaystyle{\displaystyle\mathsf{Q}^{\mu}_{\nu}\left(x\right)=\frac{\pi}{2% \sin\left(\mu\pi\right)}\left(\cos\left(\mu\pi\right)\left(\frac{1+x}{1-x}% \right)^{\mu/2}\mathbf{F}\left(\nu+1,-\nu;1-\mu;\tfrac{1}{2}-\tfrac{1}{2}x% \right)-\frac{\Gamma\left(\nu+\mu+1\right)}{\Gamma\left(\nu-\mu+1\right)}\left% (\frac{1-x}{1+x}\right)^{\mu/2}\mathbf{F}\left(\nu+1,-\nu;1+\mu;\tfrac{1}{2}-% \tfrac{1}{2}x\right)\right).}}
    0 references
    DLMF id
    DLMF:14.3.E2
    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.m2adec
    0 references
    the ratio of the circumference of a circle to its diameter
    Defining formula
    π {\displaystyle{\displaystyle\pi}}
    xml-id
    C3.S12.E1.m2adec
    0 references
    cosine function
    Defining formula
    cos ⁡ z 𝑧 {\displaystyle{\displaystyle\cos\NVar{z}}}
    xml-id
    C4.S14.E2.m2adec
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    $$={{}_{2}{\mathbf{F}}_{1}}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$$ Olver’s hypergeometric function
    Defining formula
    𝐅 ⁡ ( a , b ; c ; z ) scaled-hypergeometric-bold-F 𝑎 𝑏 𝑐 𝑧 {\displaystyle{\displaystyle\mathbf{F}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z% }\right)}}
    xml-id
    C15.S2.E2.m2aadec
    0 references
    sine function
    Defining formula
    sin ⁡ z 𝑧 {\displaystyle{\displaystyle\sin\NVar{z}}}
    xml-id
    C4.S14.E1.m2adec
    0 references
    Q11589
    Defining formula
    x 𝑥 {\displaystyle{\displaystyle x}}
    xml-id
    C14.S1.XMD1.m1adec
    0 references
    Q11591
    Defining formula
    μ 𝜇 {\displaystyle{\displaystyle\mu}}
    xml-id
    C14.S1.XMD7.m1adec
    0 references
     
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