DLMF:14.17.E12 (Q4893): Difference between revisions

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Property / Symbols used
 
Property / Symbols used: the ratio of the circumference of a circle to its diameter / rank
 
Normal rank
Property / Symbols used: the ratio of the circumference of a circle to its diameter / qualifier
 
test:

π {\displaystyle{\displaystyle\pi}}

\cpi
Property / Symbols used: the ratio of the circumference of a circle to its diameter / qualifier
 
xml-id: C3.S12.E1.m2abdec

Revision as of 12:24, 2 January 2020

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DLMF:14.17.E12
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    Statements

    test
    ∫ - 1 1 𝖰 ν ⁡ ( x ) ⁢ 𝖰 λ ⁡ ( x ) ⁢ d x = ( ( ψ ⁡ ( ν + 1 ) - ψ ⁡ ( λ + 1 ) ) ⁢ ( 1 + cos ⁡ ( ν ⁢ π ) ⁢ cos ⁡ ( λ ⁢ π ) ) + 1 2 ⁢ π ⁢ sin ⁡ ( ( λ - ν ) ⁢ π ) ) ( λ - ν ) ⁢ ( λ + ν + 1 ) , superscript subscript 1 1 shorthand-Ferrers-Legendre-Q-first-kind 𝜈 𝑥 shorthand-Ferrers-Legendre-Q-first-kind 𝜆 𝑥 𝑥 digamma 𝜈 1 digamma 𝜆 1 1 𝜈 𝜋 𝜆 𝜋 1 2 𝜋 𝜆 𝜈 𝜋 𝜆 𝜈 𝜆 𝜈 1 {\displaystyle{\displaystyle\int_{-1}^{1}\mathsf{Q}_{\nu}\left(x\right)\mathsf% {Q}_{\lambda}\left(x\right)\mathrm{d}x=\frac{\left((\psi\left(\nu+1\right)-% \psi\left(\lambda+1\right))(1+\cos\left(\nu\pi\right)\cos\left(\lambda\pi% \right))+\frac{1}{2}\pi\sin\left((\lambda-\nu)\pi\right)\right)}{(\lambda-\nu)% (\lambda+\nu+1)},}}
    0 references
    DLMF id
    DLMF:14.17.E12
    0 references
    constraint
    λ ⁢  and  ⁢ ν ≠ - 1 , - 2 , - 3 , … 𝜆  and  𝜈 1 2 3 … {\displaystyle{\displaystyle\lambda\text{ and }\nu\neq-1,-2,-3,\dots}}
    0 references
    Symbols used
    the ratio of the circumference of a circle to its diameter
    test
    π {\displaystyle{\displaystyle\pi}}
    xml-id
    C3.S12.E1.m2abdec
    0 references
     
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