DLMF:14.17.E14 (Q4895): Difference between revisions

From DRMF
Jump to navigation Jump to search
← Older editNewer edit →
‎Changed an Item: Add constraint
‎Changed an Item: Add constraint
Property / Symbols used
 
Property / Symbols used: sine function / rank
 
Normal rank
Property / Symbols used: sine function / qualifier
 
Defining formula:

sin ⁡ z 𝑧 {\displaystyle{\displaystyle\sin\NVar{z}}}

\sin@@{\NVar{z}}
Property / Symbols used: sine function / qualifier
 
xml-id: C4.S14.E1.m2acdec

Revision as of 13:25, 2 January 2020

No description defined
Language Label Description Also known as
English
DLMF:14.17.E14
No description defined

    Statements

    Defining formula
    ∫ - 1 1 𝖯 ν ⁡ ( x ) ⁢ 𝖰 λ ⁡ ( x ) ⁢ d x = 2 ⁢ sin ⁡ ( ν ⁢ π ) ⁢ cos ⁡ ( λ ⁢ π ) ⁢ ( ψ ⁡ ( ν + 1 ) - ψ ⁡ ( λ + 1 ) ) + π ⁢ cos ⁡ ( ( λ - ν ) ⁢ π ) - π π ⁢ ( λ - ν ) ⁢ ( λ + ν + 1 ) , superscript subscript 1 1 shorthand-Ferrers-Legendre-P-first-kind 𝜈 𝑥 shorthand-Ferrers-Legendre-Q-first-kind 𝜆 𝑥 𝑥 2 𝜈 𝜋 𝜆 𝜋 digamma 𝜈 1 digamma 𝜆 1 𝜋 𝜆 𝜈 𝜋 𝜋 𝜋 𝜆 𝜈 𝜆 𝜈 1 {\displaystyle{\displaystyle\int_{-1}^{1}\mathsf{P}_{\nu}\left(x\right)\mathsf% {Q}_{\lambda}\left(x\right)\mathrm{d}x=\frac{2\sin\left(\nu\pi\right)\cos\left% (\lambda\pi\right)\left(\psi\left(\nu+1\right)-\psi\left(\lambda+1\right)% \right)+\pi\cos\left((\lambda-\nu)\pi\right)-\pi}{\pi(\lambda-\nu)(\lambda+\nu% +1)},}}
    0 references
    DLMF id
    DLMF:14.17.E14
    0 references
    constraint
    ℜ ⁡ λ > 0 𝜆 0 {\displaystyle{\displaystyle\Re\lambda>0}}
    0 references
    ℜ ⁡ ν > 0 𝜈 0 {\displaystyle{\displaystyle\Re\nu>0}}
    0 references
    ℜ ⁡ λ > 0 𝜆 0 {\displaystyle{\displaystyle\Re\lambda>0}}
    0 references
    ℜ ⁡ ν > 0 𝜈 0 {\displaystyle{\displaystyle\Re\nu>0}}
    0 references
    λ ≠ ν 𝜆 𝜈 {\displaystyle{\displaystyle\lambda\neq\nu}}
    0 references
    Symbols used
    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
    cosine function
    Defining formula
    cos ⁡ z 𝑧 {\displaystyle{\displaystyle\cos\NVar{z}}}
    xml-id
    C4.S14.E2.m2abdec
    0 references
    Q10770
    Defining formula
    d x 𝑥 {\displaystyle{\displaystyle\mathrm{d}\NVar{x}}}
    xml-id
    C1.S4.SS4.m1amdec
    0 references
    psi (or digamma) function
    Defining formula
    ψ ⁡ ( z ) digamma 𝑧 {\displaystyle{\displaystyle\psi\left(\NVar{z}\right)}}
    xml-id
    C5.S2.E2.m2addec
    0 references
    Q10771
    Defining formula
    ∫ {\displaystyle{\displaystyle\int}}
    xml-id
    C1.S4.SS4.m3amdec
    0 references
    Q10811
    Defining formula
    ℜ ⁡ absent {\displaystyle{\displaystyle\Re}}
    xml-id
    C1.S9.E2.m1aadec
    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.m2abdec
    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.m4abdec
    0 references
    sine function
    Defining formula
    sin ⁡ z 𝑧 {\displaystyle{\displaystyle\sin\NVar{z}}}
    xml-id
    C4.S14.E1.m2acdec
    0 references
     
        Retrieved from "https://drmf-beta.wmflabs.org/index.php?title=Item:Q4895&oldid=36286"