DLMF:14.11.E2 (Q4818): Difference between revisions

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Property / Symbols used
 
Property / Symbols used: Q11605 / rank
 
Normal rank
Property / Symbols used: Q11605 / qualifier
 
Defining formula:

𝖠 ν μ ⁢ ( x ) superscript subscript 𝖠 𝜈 𝜇 𝑥 {\displaystyle{\displaystyle\mathsf{A}_{\nu}^{\mu}(x)}}

\mathsf{A}_{\nu}^{\mu}(x)
Property / Symbols used: Q11605 / qualifier
 
xml-id: C14.S11.XMD1.m1adec

Latest revision as of 01:40, 2 January 2020

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DLMF:14.11.E2
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    Statements

    Defining formula
    ∂ ∂ ⁡ ν ⁡ 𝖰 ν μ ⁡ ( x ) = - 1 2 ⁢ π 2 ⁢ 𝖯 ν μ ⁡ ( x ) + π ⁢ sin ⁡ ( μ ⁢ π ) sin ⁡ ( ν ⁢ π ) ⁢ sin ⁡ ( ( ν + μ ) ⁢ π ) ⁢ 𝖰 ν μ ⁡ ( x ) - 1 2 ⁢ cot ⁡ ( ( ν + μ ) ⁢ π ) ⁢ 𝖠 ν μ ⁢ ( x ) + 1 2 ⁢ csc ⁡ ( ( ν + μ ) ⁢ π ) ⁢ 𝖠 ν μ ⁢ ( - x ) , partial-derivative 𝜈 Ferrers-Legendre-Q-first-kind 𝜇 𝜈 𝑥 1 2 superscript 𝜋 2 Ferrers-Legendre-P-first-kind 𝜇 𝜈 𝑥 𝜋 𝜇 𝜋 𝜈 𝜋 𝜈 𝜇 𝜋 Ferrers-Legendre-Q-first-kind 𝜇 𝜈 𝑥 1 2 𝜈 𝜇 𝜋 superscript subscript 𝖠 𝜈 𝜇 𝑥 1 2 𝜈 𝜇 𝜋 superscript subscript 𝖠 𝜈 𝜇 𝑥 {\displaystyle{\displaystyle\frac{\partial}{\partial\nu}\mathsf{Q}^{\mu}_{\nu}% \left(x\right)=-\tfrac{1}{2}\pi^{2}\mathsf{P}^{\mu}_{\nu}\left(x\right)+\frac{% \pi\sin\left(\mu\pi\right)}{\sin\left(\nu\pi\right)\sin\left((\nu+\mu)\pi% \right)}\mathsf{Q}^{\mu}_{\nu}\left(x\right)-\tfrac{1}{2}\cot\left((\nu+\mu)% \pi\right)\mathsf{A}_{\nu}^{\mu}(x)+\tfrac{1}{2}\csc\left((\nu+\mu)\pi\right)% \mathsf{A}_{\nu}^{\mu}(-x),}}
    0 references
    DLMF id
    DLMF:14.11.E2
    0 references
    Symbols used
    Ferrers function of the first kind
    Defining formula
    𝖯 ν μ ⁡ ( x ) Ferrers-Legendre-P-first-kind 𝜇 𝜈 𝑥 {\displaystyle{\displaystyle\mathsf{P}^{\NVar{\mu}}_{\NVar{\nu}}\left(\NVar{x}% \right)}}
    xml-id
    C14.S3.E1.m2aadec
    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.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.m2aadec
    0 references
    cosecant function
    Defining formula
    csc ⁡ z 𝑧 {\displaystyle{\displaystyle\csc\NVar{z}}}
    xml-id
    C4.S14.E5.m2adec
    0 references
    cotangent function
    Defining formula
    cot ⁡ z 𝑧 {\displaystyle{\displaystyle\cot\NVar{z}}}
    xml-id
    C4.S14.E7.m2aadec
    0 references
    partial derivative of $$f$$ with respect to $$x$$
    Defining formula
    ∂ ⁡ f ∂ ⁡ x partial-derivative 𝑓 𝑥 {\displaystyle{\displaystyle\frac{\partial\NVar{f}}{\partial\NVar{x}}}}
    xml-id
    C1.S5.E3.m4aadec
    0 references
    partial differential of $$x$$
    Defining formula
    ∂ ⁡ x 𝑥 {\displaystyle{\displaystyle\partial\NVar{x}}}
    xml-id
    C1.S5.E3.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
    Q11590
    Defining formula
    ν 𝜈 {\displaystyle{\displaystyle\nu}}
    xml-id
    C14.S1.XMD8.m1adec
    0 references
    Q11605
    Defining formula
    𝖠 ν μ ⁢ ( x ) superscript subscript 𝖠 𝜈 𝜇 𝑥 {\displaystyle{\displaystyle\mathsf{A}_{\nu}^{\mu}(x)}}
    xml-id
    C14.S11.XMD1.m1adec
    0 references
     
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