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

From testwiki
Jump to navigation Jump to search
imported>Admin
 
imported>Admin
 
(9 intermediate revisions by the same user not shown)
Property / constraint
 

λ ⁢  and  ⁢ ν ≠ - 1 , - 2 , - 3 , … 𝜆  and  𝜈 1 2 3 … {\displaystyle{\displaystyle\lambda\text{ and }\nu\neq-1,-2,-3,\dots}}

\lambda\text{ and }\nu\neq-1,-2,-3,\dots
Property / constraint: λ ⁢  and  ⁢ ν ≠ - 1 , - 2 , - 3 , … 𝜆  and  𝜈 1 2 3 … {\displaystyle{\displaystyle\lambda\text{ and }\nu\neq-1,-2,-3,\dots}} / rank
 
Normal rank
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
Property / Symbols used
 
Property / Symbols used: cosine function / rank
 
Normal rank
Property / Symbols used: cosine function / qualifier
 
test:

cos ⁡ z 𝑧 {\displaystyle{\displaystyle\cos\NVar{z}}}

\cos@@{\NVar{z}}
Property / Symbols used: cosine function / qualifier
 
xml-id: C4.S14.E2.m2adec
Property / Symbols used
 
Property / Symbols used: Q10770 / rank
 
Normal rank
Property / Symbols used: Q10770 / qualifier
 
test:

d x 𝑥 {\displaystyle{\displaystyle\mathrm{d}\NVar{x}}}

\diff{\NVar{x}}
Property / Symbols used: Q10770 / qualifier
 
xml-id: C1.S4.SS4.m1akdec
Property / Symbols used
 
Property / Symbols used: psi (or digamma) function / rank
 
Normal rank
Property / Symbols used: psi (or digamma) function / qualifier
 
test:

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: Q10771 / rank
 
Normal rank
Property / Symbols used: Q10771 / qualifier
 
test:

∫ {\displaystyle{\displaystyle\int}}

\int
Property / Symbols used: Q10771 / qualifier
 
xml-id: C1.S4.SS4.m3akdec
Property / Symbols used
 
Property / Symbols used: Q11596 / rank
 
Normal rank
Property / Symbols used: Q11596 / qualifier
 
test:

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: sine function / rank
 
Normal rank
Property / Symbols used: sine function / qualifier
 
test:

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

\sin@@{\NVar{z}}
Property / Symbols used: sine function / qualifier
 
xml-id: C4.S14.E1.m2abdec
Property / Symbols used
 
Property / Symbols used: Q11589 / rank
 
Normal rank
Property / Symbols used: Q11589 / qualifier
 
test:

x 𝑥 {\displaystyle{\displaystyle x}}

x
Property / Symbols used: Q11589 / qualifier
 
xml-id: C14.S1.XMD1.m1kdec
Property / Symbols used
 
Property / Symbols used: Q11590 / rank
 
Normal rank
Property / Symbols used: Q11590 / qualifier
 
test:

ν 𝜈 {\displaystyle{\displaystyle\nu}}

\nu
Property / Symbols used: Q11590 / qualifier
 
xml-id: C14.S1.XMD8.m1gdec

Latest revision as of 12:24, 2 January 2020

No description defined
Language Label Description Also known as
English
DLMF:14.17.E12
No description defined

    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
    cosine function
    test
    cos ⁡ z 𝑧 {\displaystyle{\displaystyle\cos\NVar{z}}}
    xml-id
    C4.S14.E2.m2adec
    0 references
    Q10770
    test
    d x 𝑥 {\displaystyle{\displaystyle\mathrm{d}\NVar{x}}}
    xml-id
    C1.S4.SS4.m1akdec
    0 references
    psi (or digamma) function
    test
    ψ ⁡ ( z ) digamma 𝑧 {\displaystyle{\displaystyle\psi\left(\NVar{z}\right)}}
    xml-id
    C5.S2.E2.m2abdec
    0 references
    Q10771
    test
    ∫ {\displaystyle{\displaystyle\int}}
    xml-id
    C1.S4.SS4.m3akdec
    0 references
    Q11596
    test
    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
    sine function
    test
    sin ⁡ z 𝑧 {\displaystyle{\displaystyle\sin\NVar{z}}}
    xml-id
    C4.S14.E1.m2abdec
    0 references
    Q11589
    test
    x 𝑥 {\displaystyle{\displaystyle x}}
    xml-id
    C14.S1.XMD1.m1kdec
    0 references
    Q11590
    test
    ν 𝜈 {\displaystyle{\displaystyle\nu}}
    xml-id
    C14.S1.XMD8.m1gdec
    0 references
     
      Retrieved from "https://lct.wmflabs.org/w/index.php?title=Item:Q4893&oldid=27503"