DLMF:19.29.E8 (Q6630): Difference between revisions

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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.m3abdec
Property / Symbols used
 
Property / Symbols used: Q10777 / rank
 
Normal rank
Property / Symbols used: Q10777 / qualifier
 
test:

( a , b ) 𝑎 𝑏 {\displaystyle{\displaystyle(\NVar{a},\NVar{b})}}

(\NVar{a},\NVar{b})
Property / Symbols used: Q10777 / qualifier
 
xml-id: introduction.Sx4.p1.t1.r29.m6adec
Property / Symbols used
 
Property / Symbols used: Q10785 / rank
 
Normal rank
Property / Symbols used: Q10785 / qualifier
 
test:

∖ {\displaystyle{\displaystyle\setminus}}

\setminus
Property / Symbols used: Q10785 / qualifier
 
xml-id: introduction.Sx4.p2.t1.r19.m2adec
Property / Symbols used
 
Property / Symbols used: Q11899 / rank
 
Normal rank
Property / Symbols used: Q11899 / qualifier
 
test:

U n subscript 𝑈 𝑛 {\displaystyle{\displaystyle U_{n}}}

U_{n}
Property / Symbols used: Q11899 / qualifier
 
xml-id: C19.S29.XMD10.m1ddec
Property / Symbols used
 
Property / Symbols used: Q11904 / rank
 
Normal rank
Property / Symbols used: Q11904 / qualifier
 
test:

S n subscript 𝑆 𝑛 {\displaystyle{\displaystyle S_{n}}}

S_{n}
Property / Symbols used: Q11904 / qualifier
 
xml-id: C19.S29.XMD11.m1dec
Property / Symbols used
 
Property / Symbols used: Q11905 / rank
 
Normal rank
Property / Symbols used: Q11905 / qualifier
 
test:

Q α ⁢ β subscript 𝑄 𝛼 𝛽 {\displaystyle{\displaystyle Q_{\alpha\beta}}}

Q_{\alpha\beta}
Property / Symbols used: Q11905 / qualifier
 
xml-id: C19.S29.XMD12.m1dec
Property / Symbols used
 
Property / Symbols used: Q11895 / rank
 
Normal rank
Property / Symbols used: Q11895 / qualifier
 
test:

d α ⁢ β subscript 𝑑 𝛼 𝛽 {\displaystyle{\displaystyle d_{\alpha\beta}}}

d_{\alpha\beta}
Property / Symbols used: Q11895 / qualifier
 
xml-id: C19.S29.XMD3.m1cdec
Property / Symbols used
 
Property / Symbols used: Q11898 / rank
 
Normal rank
Property / Symbols used: Q11898 / qualifier
 
test:

s ⁢ ( t ) 𝑠 𝑡 {\displaystyle{\displaystyle s(t)}}

s(t)
Property / Symbols used: Q11898 / qualifier
 
xml-id: C19.S29.XMD4.m1cdec
Property / Symbols used
 
Property / Symbols used: Q11896 / rank
 
Normal rank
Property / Symbols used: Q11896 / qualifier
 
test:

α 𝛼 {\displaystyle{\displaystyle\alpha}}

\alpha
Property / Symbols used: Q11896 / qualifier
 
xml-id: C19.S29.XMD5.m1fdec
Property / Symbols used
 
Property / Symbols used: Q11897 / rank
 
Normal rank
Property / Symbols used: Q11897 / qualifier
 
test:

β 𝛽 {\displaystyle{\displaystyle\beta}}

\beta
Property / Symbols used: Q11897 / qualifier
 
xml-id: C19.S29.XMD6.m1ddec
Property / Symbols used
 
Property / Symbols used: Q11902 / rank
 
Normal rank
Property / Symbols used: Q11902 / qualifier
 
test:

γ 𝛾 {\displaystyle{\displaystyle\gamma}}

\gamma
Property / Symbols used: Q11902 / qualifier
 
xml-id: C19.S29.XMD7.m1cdec
Property / Symbols used
 
Property / Symbols used: Q11903 / rank
 
Normal rank
Property / Symbols used: Q11903 / qualifier
 
test:

δ 𝛿 {\displaystyle{\displaystyle\delta}}

\delta
Property / Symbols used: Q11903 / qualifier
 
xml-id: C19.S29.XMD8.m1cdec

Latest revision as of 13:19, 2 January 2020

No description defined
Language Label Description Also known as
English
DLMF:19.29.E8
No description defined

    Statements

    test
    ∫ y x a α + b α ⁢ t a 5 + b 5 ⁢ t ⁢ d t s ⁢ ( t ) = 2 3 ⁢ d α ⁢ β ⁢ d α ⁢ γ ⁢ d α ⁢ δ d α ⁢ 5 ⁢ R J ⁡ ( U 12 2 , U 13 2 , U 23 2 , U α ⁢ 5 2 ) + 2 ⁢ R C ⁡ ( S α ⁢ 5 2 , Q α ⁢ 5 2 ) , superscript subscript 𝑦 𝑥 subscript 𝑎 𝛼 subscript 𝑏 𝛼 𝑡 subscript 𝑎 5 subscript 𝑏 5 𝑡 𝑡 𝑠 𝑡 2 3 subscript 𝑑 𝛼 𝛽 subscript 𝑑 𝛼 𝛾 subscript 𝑑 𝛼 𝛿 subscript 𝑑 𝛼 5 Carlson-integral-RJ superscript subscript 𝑈 12 2 superscript subscript 𝑈 13 2 superscript subscript 𝑈 23 2 superscript subscript 𝑈 𝛼 5 2 2 Carlson-integral-RC superscript subscript 𝑆 𝛼 5 2 superscript subscript 𝑄 𝛼 5 2 {\displaystyle{\displaystyle\int_{y}^{x}\frac{a_{\alpha}+b_{\alpha}t}{a_{5}+b_% {5}t}\frac{\mathrm{d}t}{s(t)}=\frac{2}{3}\frac{d_{\alpha\beta}d_{\alpha\gamma}% d_{\alpha\delta}}{d_{\alpha 5}}R_{J}\left(U_{12}^{2},U_{13}^{2},U_{23}^{2},U_{% \alpha 5}^{2}\right)+2R_{C}\left(S_{\alpha 5}^{2},Q_{\alpha 5}^{2}\right),}}
    0 references
    DLMF id
    DLMF:19.29.E8
    0 references
    constraint
    S α ⁢ 5 2 ∈ ℂ ∖ ( - ∞ , 0 ) superscript subscript 𝑆 𝛼 5 2 0 {\displaystyle{\displaystyle S_{\alpha 5}^{2}\in\mathbb{C}\setminus(-\infty,0)}}
    0 references
    Symbols used
    Carlson’s combination of inverse circular and inverse hyperbolic functions
    test
    R C ⁡ ( x , y ) Carlson-integral-RC 𝑥 𝑦 {\displaystyle{\displaystyle R_{C}\left(\NVar{x},\NVar{y}\right)}}
    xml-id
    C19.S2.E17.m2adec
    0 references
    symmetric elliptic integral of third kind
    test
    R J ⁡ ( x , y , z , p ) Carlson-integral-RJ 𝑥 𝑦 𝑧 𝑝 {\displaystyle{\displaystyle R_{J}\left(\NVar{x},\NVar{y},\NVar{z},\NVar{p}% \right)}}
    xml-id
    C19.S16.E2.m2adec
    0 references
    Q10923
    test
    ℂ {\displaystyle{\displaystyle\mathbb{C}}}
    xml-id
    introduction.Sx4.p1.t1.r1.m2adec
    0 references
    Q10770
    test
    d x 𝑥 {\displaystyle{\displaystyle\mathrm{d}\NVar{x}}}
    xml-id
    C1.S4.SS4.m1abdec
    0 references
    Q10784
    test
    ∈ {\displaystyle{\displaystyle\in}}
    xml-id
    introduction.Sx4.p1.t1.r10.m2adec
    0 references
    Q10771
    test
    ∫ {\displaystyle{\displaystyle\int}}
    xml-id
    C1.S4.SS4.m3abdec
    0 references
    Q10777
    test
    ( a , b ) 𝑎 𝑏 {\displaystyle{\displaystyle(\NVar{a},\NVar{b})}}
    xml-id
    introduction.Sx4.p1.t1.r29.m6adec
    0 references
    Q10785
    test
    ∖ {\displaystyle{\displaystyle\setminus}}
    xml-id
    introduction.Sx4.p2.t1.r19.m2adec
    0 references
    Q11899
    test
    U n subscript 𝑈 𝑛 {\displaystyle{\displaystyle U_{n}}}
    xml-id
    C19.S29.XMD10.m1ddec
    0 references
    Q11904
    test
    S n subscript 𝑆 𝑛 {\displaystyle{\displaystyle S_{n}}}
    xml-id
    C19.S29.XMD11.m1dec
    0 references
    Q11905
    test
    Q α ⁢ β subscript 𝑄 𝛼 𝛽 {\displaystyle{\displaystyle Q_{\alpha\beta}}}
    xml-id
    C19.S29.XMD12.m1dec
    0 references
    Q11895
    test
    d α ⁢ β subscript 𝑑 𝛼 𝛽 {\displaystyle{\displaystyle d_{\alpha\beta}}}
    xml-id
    C19.S29.XMD3.m1cdec
    0 references
    Q11898
    test
    s ⁢ ( t ) 𝑠 𝑡 {\displaystyle{\displaystyle s(t)}}
    xml-id
    C19.S29.XMD4.m1cdec
    0 references
    Q11896
    test
    α 𝛼 {\displaystyle{\displaystyle\alpha}}
    xml-id
    C19.S29.XMD5.m1fdec
    0 references
    Q11897
    test
    β 𝛽 {\displaystyle{\displaystyle\beta}}
    xml-id
    C19.S29.XMD6.m1ddec
    0 references
    Q11902
    test
    γ 𝛾 {\displaystyle{\displaystyle\gamma}}
    xml-id
    C19.S29.XMD7.m1cdec
    0 references
    Q11903
    test
    δ 𝛿 {\displaystyle{\displaystyle\delta}}
    xml-id
    C19.S29.XMD8.m1cdec
    0 references
     
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