DLMF:14.11.E5 (Q4821): Difference between revisions

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Property / Symbols used
 
Property / Symbols used: cotangent function / rank
 
Normal rank
Property / Symbols used: cotangent function / qualifier
 
Defining formula:

cot ⁡ z 𝑧 {\displaystyle{\displaystyle\cot\NVar{z}}}

\cot@@{\NVar{z}}
Property / Symbols used: cotangent function / qualifier
 
xml-id: C4.S14.E7.m2acdec
Property / Symbols used
 
Property / Symbols used: psi (or digamma) function / rank
 
Normal rank
Property / Symbols used: psi (or digamma) function / qualifier
 
Defining formula:

ψ ⁡ ( z ) digamma 𝑧 {\displaystyle{\displaystyle\psi\left(\NVar{z}\right)}}

\digamma@{\NVar{z}}
Property / Symbols used: psi (or digamma) function / qualifier
 
xml-id: C5.S2.E2.m2abdec
Property / Symbols used
 
Property / Symbols used: partial derivative of $$f$$ with respect to $$x$$ / rank
 
Normal rank
Property / Symbols used: partial derivative of $$f$$ with respect to $$x$$ / qualifier
 
Defining formula:

∂ ⁡ f ∂ ⁡ x partial-derivative 𝑓 𝑥 {\displaystyle{\displaystyle\frac{\partial\NVar{f}}{\partial\NVar{x}}}}

\pderiv{\NVar{f}}{\NVar{x}}
Property / Symbols used: partial derivative of $$f$$ with respect to $$x$$ / qualifier
 
xml-id: C1.S5.E3.m4acdec
Property / Symbols used
 
Property / Symbols used: partial differential of $$x$$ / rank
 
Normal rank
Property / Symbols used: partial differential of $$x$$ / qualifier
 
Defining formula:

∂ ⁡ x 𝑥 {\displaystyle{\displaystyle\partial\NVar{x}}}

\pdiff{\NVar{x}}
Property / Symbols used: partial differential of $$x$$ / qualifier
 
xml-id: C1.S5.E3.m2acdec
Property / Symbols used
 
Property / Symbols used: Q11595 / rank
 
Normal rank
Property / Symbols used: Q11595 / qualifier
 
Defining formula:

𝖯 ν ⁡ ( x ) = 𝖯 ν 0 ⁡ ( x ) shorthand-Ferrers-Legendre-P-first-kind 𝜈 𝑥 Ferrers-Legendre-P-first-kind 0 𝜈 𝑥 {\displaystyle{\displaystyle\mathsf{P}_{\NVar{\nu}}\left(\NVar{x}\right)=% \mathsf{P}^{0}_{\nu}\left(x\right)}}

\FerrersP[]{\NVar{\nu}}@{\NVar{x}}=\FerrersP[0]{\nu}@{x}
Property / Symbols used: Q11595 / qualifier
 
xml-id: C14.S2.SS2.p2.m2aadec
Property / Symbols used
 
Property / Symbols used: Q11596 / rank
 
Normal rank
Property / Symbols used: Q11596 / qualifier
 
Defining formula:

𝖰 ν ⁡ ( x ) = 𝖰 ν 0 ⁡ ( x ) shorthand-Ferrers-Legendre-Q-first-kind 𝜈 𝑥 Ferrers-Legendre-Q-first-kind 0 𝜈 𝑥 {\displaystyle{\displaystyle\mathsf{Q}_{\NVar{\nu}}\left(\NVar{x}\right)=% \mathsf{Q}^{0}_{\nu}\left(x\right)}}

\FerrersQ[]{\NVar{\nu}}@{\NVar{x}}=\FerrersQ[0]{\nu}@{x}
Property / Symbols used: Q11596 / qualifier
 
xml-id: C14.S2.SS2.p2.m4aadec
Property / Symbols used
 
Property / Symbols used: Q11589 / rank
 
Normal rank
Property / Symbols used: Q11589 / qualifier
 
Defining formula:

x 𝑥 {\displaystyle{\displaystyle x}}

x
Property / Symbols used: Q11589 / qualifier
 
xml-id: C14.S1.XMD1.m1ddec
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.m1ddec
Property / Symbols used
 
Property / Symbols used: Q11590 / rank
 
Normal rank
Property / Symbols used: Q11590 / qualifier
 
Defining formula:

ν 𝜈 {\displaystyle{\displaystyle\nu}}

\nu
Property / Symbols used: Q11590 / qualifier
 
xml-id: C14.S1.XMD8.m1ddec

Latest revision as of 01:41, 2 January 2020

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DLMF:14.11.E5
No description defined

    Statements

    Defining formula
    ∂ ∂ ⁡ μ ⁡ 𝖰 ν μ ⁡ ( x ) | μ = 0 = - 1 4 ⁢ π 2 ⁢ 𝖯 ν ⁡ ( x ) + ( ψ ⁡ ( - ν ) - π ⁢ cot ⁡ ( ν ⁢ π ) ) ⁢ 𝖰 ν ⁡ ( x ) . evaluated-at partial-derivative 𝜇 Ferrers-Legendre-Q-first-kind 𝜇 𝜈 𝑥 𝜇 0 1 4 superscript 𝜋 2 shorthand-Ferrers-Legendre-P-first-kind 𝜈 𝑥 digamma 𝜈 𝜋 𝜈 𝜋 shorthand-Ferrers-Legendre-Q-first-kind 𝜈 𝑥 {\displaystyle{\displaystyle\left.\frac{\partial}{\partial\mu}\mathsf{Q}^{\mu}% _{\nu}\left(x\right)\right|_{\mu=0}=-\tfrac{1}{4}\pi^{2}\mathsf{P}_{\nu}\left(% x\right)+\left(\psi\left(-\nu\right)-\pi\cot\left(\nu\pi\right)\right)\mathsf{% Q}_{\nu}\left(x\right).}}
    0 references
    DLMF id
    DLMF:14.11.E5
    0 references
    Symbols used
    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
    the ratio of the circumference of a circle to its diameter
    Defining formula
    π {\displaystyle{\displaystyle\pi}}
    xml-id
    C3.S12.E1.m2addec
    0 references
    cotangent function
    Defining formula
    cot ⁡ z 𝑧 {\displaystyle{\displaystyle\cot\NVar{z}}}
    xml-id
    C4.S14.E7.m2acdec
    0 references
    psi (or digamma) function
    Defining formula
    ψ ⁡ ( z ) digamma 𝑧 {\displaystyle{\displaystyle\psi\left(\NVar{z}\right)}}
    xml-id
    C5.S2.E2.m2abdec
    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.m4acdec
    0 references
    partial differential of $$x$$
    Defining formula
    ∂ ⁡ x 𝑥 {\displaystyle{\displaystyle\partial\NVar{x}}}
    xml-id
    C1.S5.E3.m2acdec
    0 references
    Q11595
    Defining formula
    𝖯 ν ⁡ ( x ) = 𝖯 ν 0 ⁡ ( x ) shorthand-Ferrers-Legendre-P-first-kind 𝜈 𝑥 Ferrers-Legendre-P-first-kind 0 𝜈 𝑥 {\displaystyle{\displaystyle\mathsf{P}_{\NVar{\nu}}\left(\NVar{x}\right)=% \mathsf{P}^{0}_{\nu}\left(x\right)}}
    xml-id
    C14.S2.SS2.p2.m2aadec
    0 references
    Q11596
    Defining formula
    𝖰 ν ⁡ ( x ) = 𝖰 ν 0 ⁡ ( x ) shorthand-Ferrers-Legendre-Q-first-kind 𝜈 𝑥 Ferrers-Legendre-Q-first-kind 0 𝜈 𝑥 {\displaystyle{\displaystyle\mathsf{Q}_{\NVar{\nu}}\left(\NVar{x}\right)=% \mathsf{Q}^{0}_{\nu}\left(x\right)}}
    xml-id
    C14.S2.SS2.p2.m4aadec
    0 references
    Q11589
    Defining formula
    x 𝑥 {\displaystyle{\displaystyle x}}
    xml-id
    C14.S1.XMD1.m1ddec
    0 references
    Q11591
    Defining formula
    μ 𝜇 {\displaystyle{\displaystyle\mu}}
    xml-id
    C14.S1.XMD7.m1ddec
    0 references
    Q11590
    Defining formula
    ν 𝜈 {\displaystyle{\displaystyle\nu}}
    xml-id
    C14.S1.XMD8.m1ddec
    0 references
     
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