DLMF:14.7.E2 (Q4752): Difference between revisions

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Property / Symbols used
 
Property / Symbols used: Q10920 / rank
 
Normal rank
Property / Symbols used: Q10920 / qualifier
 
Defining formula:

P n ⁡ ( x ) Legendre-spherical-polynomial 𝑛 𝑥 {\displaystyle{\displaystyle P_{\NVar{n}}\left(\NVar{x}\right)}}

\LegendrepolyP{\NVar{n}}@{\NVar{x}}
Property / Symbols used: Q10920 / qualifier
 
xml-id: C18.S3.T1.t1.r10.m2aadec
Property / Symbols used
 
Property / Symbols used: principal branch of logarithm function / rank
 
Normal rank
Property / Symbols used: principal branch of logarithm function / qualifier
 
Defining formula:

ln ⁡ z 𝑧 {\displaystyle{\displaystyle\ln\NVar{z}}}

\ln@@{\NVar{z}}
Property / Symbols used: principal branch of logarithm function / qualifier
 
xml-id: C4.S2.E2.m2adec
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.m4adec
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.m1adec
Property / Symbols used
 
Property / Symbols used: Q11602 / rank
 
Normal rank
Property / Symbols used: Q11602 / qualifier
 
Defining formula:

n 𝑛 {\displaystyle{\displaystyle n}}

n
Property / Symbols used: Q11602 / qualifier
 
xml-id: C14.S1.XMD6.m1adec
Property / Symbols used
 
Property / Symbols used: Q11603 / rank
 
Normal rank
Property / Symbols used: Q11603 / qualifier
 
Defining formula:

W n - 1 ⁢ ( x ) subscript 𝑊 𝑛 1 𝑥 {\displaystyle{\displaystyle W_{n-1}(x)}}

W_{n-1}(x)
Property / Symbols used: Q11603 / qualifier
 
xml-id: C14.S7.XMD1.m1dec

Latest revision as of 01:28, 2 January 2020

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DLMF:14.7.E2
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    Statements

    Defining formula
    𝖰 n 0 ⁡ ( x ) = 𝖰 n ⁡ ( x ) = 1 2 ⁢ P n ⁡ ( x ) ⁢ ln ⁡ ( 1 + x 1 - x ) - W n - 1 ⁢ ( x ) , Ferrers-Legendre-Q-first-kind 0 𝑛 𝑥 shorthand-Ferrers-Legendre-Q-first-kind 𝑛 𝑥 1 2 Legendre-spherical-polynomial 𝑛 𝑥 1 𝑥 1 𝑥 subscript 𝑊 𝑛 1 𝑥 {\displaystyle{\displaystyle\mathsf{Q}^{0}_{n}\left(x\right)=\mathsf{Q}_{n}% \left(x\right)=\frac{1}{2}P_{n}\left(x\right)\ln\left(\frac{1+x}{1-x}\right)-W% _{n-1}(x),}}
    0 references
    DLMF id
    DLMF:14.7.E2
    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.m2adec
    0 references
    Q10920
    Defining formula
    P n ⁡ ( x ) Legendre-spherical-polynomial 𝑛 𝑥 {\displaystyle{\displaystyle P_{\NVar{n}}\left(\NVar{x}\right)}}
    xml-id
    C18.S3.T1.t1.r10.m2aadec
    0 references
    principal branch of logarithm function
    Defining formula
    ln ⁡ z 𝑧 {\displaystyle{\displaystyle\ln\NVar{z}}}
    xml-id
    C4.S2.E2.m2adec
    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.m4adec
    0 references
    Q11589
    Defining formula
    x 𝑥 {\displaystyle{\displaystyle x}}
    xml-id
    C14.S1.XMD1.m1adec
    0 references
    Q11602
    Defining formula
    n 𝑛 {\displaystyle{\displaystyle n}}
    xml-id
    C14.S1.XMD6.m1adec
    0 references
    Q11603
    Defining formula
    W n - 1 ⁢ ( x ) subscript 𝑊 𝑛 1 𝑥 {\displaystyle{\displaystyle W_{n-1}(x)}}
    xml-id
    C14.S7.XMD1.m1dec
    0 references
     
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