DLMF:19.29.E7 (Q6629): Difference between revisions

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Property / Symbols used
 
Property / Symbols used: Q11896 / rank
 
Normal rank
Property / Symbols used: Q11896 / qualifier
 
Defining formula:

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

\alpha
Property / Symbols used: Q11896 / qualifier
 
xml-id: C19.S29.XMD5.m1edec
Property / Symbols used
 
Property / Symbols used: Q11897 / rank
 
Normal rank
Property / Symbols used: Q11897 / qualifier
 
Defining formula:

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

\beta
Property / Symbols used: Q11897 / qualifier
 
xml-id: C19.S29.XMD6.m1cdec
Property / Symbols used
 
Property / Symbols used: Q11902 / rank
 
Normal rank
Property / Symbols used: Q11902 / qualifier
 
Defining formula:

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

\gamma
Property / Symbols used: Q11902 / qualifier
 
xml-id: C19.S29.XMD7.m1bdec
Property / Symbols used
 
Property / Symbols used: Q11903 / rank
 
Normal rank
Property / Symbols used: Q11903 / qualifier
 
Defining formula:

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

\delta
Property / Symbols used: Q11903 / qualifier
 
xml-id: C19.S29.XMD8.m1bdec

Latest revision as of 14:19, 2 January 2020

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DLMF:19.29.E7
No description defined

    Statements

    Defining formula
    ∫ y x a α + b α ⁢ t a δ + b δ ⁢ t ⁢ d t s ⁢ ( t ) = 2 3 ⁢ d α ⁢ β ⁢ d α ⁢ γ ⁢ R D ⁡ ( U α ⁢ β 2 , U α ⁢ γ 2 , U α ⁢ δ 2 ) + 2 ⁢ X α ⁢ Y α X δ ⁢ Y δ ⁢ U α ⁢ δ , superscript subscript 𝑦 𝑥 subscript 𝑎 𝛼 subscript 𝑏 𝛼 𝑡 subscript 𝑎 𝛿 subscript 𝑏 𝛿 𝑡 𝑡 𝑠 𝑡 2 3 subscript 𝑑 𝛼 𝛽 subscript 𝑑 𝛼 𝛾 Carlson-integral-RD superscript subscript 𝑈 𝛼 𝛽 2 superscript subscript 𝑈 𝛼 𝛾 2 superscript subscript 𝑈 𝛼 𝛿 2 2 subscript 𝑋 𝛼 subscript 𝑌 𝛼 subscript 𝑋 𝛿 subscript 𝑌 𝛿 subscript 𝑈 𝛼 𝛿 {\displaystyle{\displaystyle\int_{y}^{x}\frac{a_{\alpha}+b_{\alpha}t}{a_{% \delta}+b_{\delta}t}\frac{\mathrm{d}t}{s(t)}=\tfrac{2}{3}d_{\alpha\beta}d_{% \alpha\gamma}R_{D}\left(U_{\alpha\beta}^{2},U_{\alpha\gamma}^{2},U_{\alpha% \delta}^{2}\right)+\frac{2X_{\alpha}Y_{\alpha}}{X_{\delta}Y_{\delta}U_{\alpha% \delta}},}}
    0 references
    DLMF id
    DLMF:19.29.E7
    0 references
    constraint
    U α ⁢ δ ≠ 0 subscript 𝑈 𝛼 𝛿 0 {\displaystyle{\displaystyle U_{\alpha\delta}\neq 0}}
    0 references
    Symbols used
    elliptic integral symmetric in only two variables
    Defining formula
    R D ⁡ ( x , y , z ) Carlson-integral-RD 𝑥 𝑦 𝑧 {\displaystyle{\displaystyle R_{D}\left(\NVar{x},\NVar{y},\NVar{z}\right)}}
    xml-id
    C19.S16.E5.m2adec
    0 references
    Q10770
    Defining formula
    d x 𝑥 {\displaystyle{\displaystyle\mathrm{d}\NVar{x}}}
    xml-id
    C1.S4.SS4.m1aadec
    0 references
    Q10771
    Defining formula
    ∫ {\displaystyle{\displaystyle\int}}
    xml-id
    C1.S4.SS4.m3aadec
    0 references
    Q11900
    Defining formula
    X α subscript 𝑋 𝛼 {\displaystyle{\displaystyle X_{\alpha}}}
    xml-id
    C19.S29.XMD1.m1cdec
    0 references
    Q11899
    Defining formula
    U n subscript 𝑈 𝑛 {\displaystyle{\displaystyle U_{n}}}
    xml-id
    C19.S29.XMD10.m1cdec
    0 references
    Q11901
    Defining formula
    Y α subscript 𝑌 𝛼 {\displaystyle{\displaystyle Y_{\alpha}}}
    xml-id
    C19.S29.XMD2.m1cdec
    0 references
    Q11895
    Defining formula
    d α ⁢ β subscript 𝑑 𝛼 𝛽 {\displaystyle{\displaystyle d_{\alpha\beta}}}
    xml-id
    C19.S29.XMD3.m1bdec
    0 references
    Q11898
    Defining formula
    s ⁢ ( t ) 𝑠 𝑡 {\displaystyle{\displaystyle s(t)}}
    xml-id
    C19.S29.XMD4.m1bdec
    0 references
    Q11896
    Defining formula
    α 𝛼 {\displaystyle{\displaystyle\alpha}}
    xml-id
    C19.S29.XMD5.m1edec
    0 references
    Q11897
    Defining formula
    β 𝛽 {\displaystyle{\displaystyle\beta}}
    xml-id
    C19.S29.XMD6.m1cdec
    0 references
    Q11902
    Defining formula
    γ 𝛾 {\displaystyle{\displaystyle\gamma}}
    xml-id
    C19.S29.XMD7.m1bdec
    0 references
    Q11903
    Defining formula
    δ 𝛿 {\displaystyle{\displaystyle\delta}}
    xml-id
    C19.S29.XMD8.m1bdec
    0 references
     
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