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

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‎Changed an Item: Add constraint
‎Changed an Item: Add constraint
Property / Symbols used
 
Property / Symbols used: Q11904 / rank
 
Normal rank
Property / Symbols used: Q11904 / qualifier
 
Defining formula:

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

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

Revision as of 14:19, 2 January 2020

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DLMF:19.29.E8
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    Statements

    Defining formula
    ∫ 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
    Defining formula
    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
    Defining formula
    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
    Defining formula
    ℂ {\displaystyle{\displaystyle\mathbb{C}}}
    xml-id
    introduction.Sx4.p1.t1.r1.m2adec
    0 references
    Q10770
    Defining formula
    d x 𝑥 {\displaystyle{\displaystyle\mathrm{d}\NVar{x}}}
    xml-id
    C1.S4.SS4.m1abdec
    0 references
    Q10784
    Defining formula
    ∈ {\displaystyle{\displaystyle\in}}
    xml-id
    introduction.Sx4.p1.t1.r10.m2adec
    0 references
    Q10771
    Defining formula
    ∫ {\displaystyle{\displaystyle\int}}
    xml-id
    C1.S4.SS4.m3abdec
    0 references
    Q10777
    Defining formula
    ( a , b ) 𝑎 𝑏 {\displaystyle{\displaystyle(\NVar{a},\NVar{b})}}
    xml-id
    introduction.Sx4.p1.t1.r29.m6adec
    0 references
    Q10785
    Defining formula
    ∖ {\displaystyle{\displaystyle\setminus}}
    xml-id
    introduction.Sx4.p2.t1.r19.m2adec
    0 references
    Q11899
    Defining formula
    U n subscript 𝑈 𝑛 {\displaystyle{\displaystyle U_{n}}}
    xml-id
    C19.S29.XMD10.m1ddec
    0 references
    Q11904
    Defining formula
    S n subscript 𝑆 𝑛 {\displaystyle{\displaystyle S_{n}}}
    xml-id
    C19.S29.XMD11.m1dec
    0 references
     
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