DLMF:19.33.E7 (Q6705)

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DLMF:19.33.E7
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    Statements

    Defining formula
    L c = 2 ⁢ π ⁢ a ⁢ b ⁢ c ⁢ ∫ 0 ∞ d λ ( a 2 + λ ) ⁢ ( b 2 + λ ) ⁢ ( c 2 + λ ) 3 = V ⁢ R D ⁡ ( a 2 , b 2 , c 2 ) . subscript 𝐿 𝑐 2 𝜋 𝑎 𝑏 𝑐 superscript subscript 0 𝜆 superscript 𝑎 2 𝜆 superscript 𝑏 2 𝜆 superscript superscript 𝑐 2 𝜆 3 𝑉 Carlson-integral-RD superscript 𝑎 2 superscript 𝑏 2 superscript 𝑐 2 {\displaystyle{\displaystyle L_{c}=2\pi abc\int_{0}^{\infty}\frac{\mathrm{d}% \lambda}{\sqrt{(a^{2}+\lambda)(b^{2}+\lambda)(c^{2}+\lambda)^{3}}}=VR_{D}\left% (a^{2},b^{2},c^{2}\right).}}
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    DLMF id
    DLMF:19.33.E7
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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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    the ratio of the circumference of a circle to its diameter
    Defining formula
    π {\displaystyle{\displaystyle\pi}}
    xml-id
    C3.S12.E1.m2aadec
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    Q10770
    Defining formula
    d x 𝑥 {\displaystyle{\displaystyle\mathrm{d}\NVar{x}}}
    xml-id
    C1.S4.SS4.m1adec
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    Q10771
    Defining formula
    ∫ {\displaystyle{\displaystyle\int}}
    xml-id
    C1.S4.SS4.m3adec
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    Q11929
    Defining formula
    L a subscript 𝐿 𝑎 {\displaystyle{\displaystyle L_{a}}}
    xml-id
    C19.S33.XMD10.m1dec
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