DLMF:19.14.E4 (Q6309)

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DLMF:19.14.E4
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    Statements

    Defining formula
    ∫ y x d t ( a 1 + b 1 ⁢ t 2 ) ⁢ ( a 2 + b 2 ⁢ t 2 ) = 1 γ - α ⁢ F ⁡ ( ϕ , k ) , superscript subscript 𝑦 𝑥 𝑡 subscript 𝑎 1 subscript 𝑏 1 superscript 𝑡 2 subscript 𝑎 2 subscript 𝑏 2 superscript 𝑡 2 1 𝛾 𝛼 elliptic-integral-first-kind-F italic-ϕ 𝑘 {\displaystyle{\displaystyle{\int_{y}^{x}\frac{\mathrm{d}t}{\sqrt{(a_{1}+b_{1}% t^{2})(a_{2}+b_{2}t^{2})}}=\frac{1}{\sqrt{\gamma-\alpha}}F\left(\phi,k\right)}% ,}}
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    DLMF id
    DLMF:19.14.E4
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    constraint
    k 2 = ( γ - β ) / ( γ - α ) superscript 𝑘 2 𝛾 𝛽 𝛾 𝛼 {\displaystyle{\displaystyle k^{2}=\ifrac{(\gamma-\beta)}{(\gamma-\alpha)}}}
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    Symbols used
    Q10770
    Defining formula
    d x 𝑥 {\displaystyle{\displaystyle\mathrm{d}\NVar{x}}}
    xml-id
    C1.S4.SS4.m1acdec
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    Legendre’s incomplete elliptic integral of the first kind
    Defining formula
    F ⁡ ( ϕ , k ) elliptic-integral-first-kind-F italic-ϕ 𝑘 {\displaystyle{\displaystyle F\left(\NVar{\phi},\NVar{k}\right)}}
    xml-id
    C19.S2.E4.m2acdec
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    Q10771
    Defining formula
    ∫ {\displaystyle{\displaystyle\int}}
    xml-id
    C1.S4.SS4.m3acdec
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    Q11807
    Defining formula
    ϕ italic-ϕ {\displaystyle{\displaystyle\phi}}
    xml-id
    C19.S1.XMD4.m1cdec
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    Q11808
    Defining formula
    k 𝑘 {\displaystyle{\displaystyle k}}
    xml-id
    C19.S1.XMD5.m1cdec
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