DLMF:19.12.E5 (Q6303)

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DLMF:19.12.E5
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    Statements

    Defining formula
    ( 1 - α 2 ) ⁢ Π ⁡ ( α 2 , k ) = ( ln ⁡ ( 4 k ′ ) - R C ⁡ ( 1 , 1 - α - 2 ) ) ⁢ ( 1 + O ⁡ ( k ′ 2 ) ) , 1 superscript 𝛼 2 complete-elliptic-integral-third-kind-Pi superscript 𝛼 2 𝑘 4 superscript 𝑘 ′ Carlson-integral-RC 1 1 superscript 𝛼 2 1 Big-O superscript superscript 𝑘 ′ 2 {\displaystyle{\displaystyle(1-\alpha^{2})\Pi\left(\alpha^{2},k\right)=\left(% \ln\left(\frac{4}{k^{\prime}}\right)-R_{C}\left(1,1-\alpha^{-2}\right)\right)% \*\left(1+O\left({k^{\prime}}^{2}\right)\right),}}
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    DLMF id
    DLMF:19.12.E5
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    constraint
    1 < α 2 < ∞ 1 superscript 𝛼 2 {\displaystyle{\displaystyle 1<\alpha^{2}<\infty}}
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    Symbols used
    order not exceeding
    Defining formula
    O ⁡ ( x ) Big-O 𝑥 {\displaystyle{\displaystyle O\left(\NVar{x}\right)}}
    xml-id
    C2.S1.E3.m2aadec
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    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.m2aadec
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    Q11816
    Defining formula
    Π ⁡ ( α 2 , k ) complete-elliptic-integral-third-kind-Pi superscript 𝛼 2 𝑘 {\displaystyle{\displaystyle\Pi\left(\NVar{\alpha}^{2},\NVar{k}\right)}}
    xml-id
    C19.S2.E8.m4aadec
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    principal branch of logarithm function
    Defining formula
    ln ⁡ z 𝑧 {\displaystyle{\displaystyle\ln\NVar{z}}}
    xml-id
    C4.S2.E2.m2acdec
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    Q11808
    Defining formula
    k 𝑘 {\displaystyle{\displaystyle k}}
    xml-id
    C19.S1.XMD5.m1cdec
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