DLMF:19.29.E7 (Q6629)

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DLMF:19.29.E7
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    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}},}}
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    DLMF id
    DLMF:19.29.E7
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    constraint
    U α ⁢ δ ≠ 0 subscript 𝑈 𝛼 𝛿 0 {\displaystyle{\displaystyle U_{\alpha\delta}\neq 0}}
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    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
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    Q10770
    Defining formula
    d x 𝑥 {\displaystyle{\displaystyle\mathrm{d}\NVar{x}}}
    xml-id
    C1.S4.SS4.m1aadec
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