DLMF:19.8.E7 (Q6213)

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DLMF:19.8.E7
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    Statements

    Defining formula
    Π ⁡ ( α 2 , k ) = π 4 ⁢ M ⁡ ( 1 , k ′ ) ⁢ ( 2 + α 2 1 - α 2 ⁢ ∑ n = 0 ∞ Q n ) , complete-elliptic-integral-third-kind-Pi superscript 𝛼 2 𝑘 𝜋 4 arithmetic-geometric-mean 1 superscript 𝑘 ′ 2 superscript 𝛼 2 1 superscript 𝛼 2 superscript subscript 𝑛 0 subscript 𝑄 𝑛 {\displaystyle{\displaystyle\Pi\left(\alpha^{2},k\right)=\frac{\pi}{4M\left(1,% k^{\prime}\right)}\left(2+\frac{\alpha^{2}}{1-\alpha^{2}}\sum_{n=0}^{\infty}Q_% {n}\right),}}
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    DLMF id
    DLMF:19.8.E7
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    constraint
    - ∞ < α 2 < 1 superscript 𝛼 2 1 {\displaystyle{\displaystyle-\infty<\alpha^{2}<1}}
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    Symbols used
    Q11826
    Defining formula
    M ⁡ ( a , g ) arithmetic-geometric-mean 𝑎 𝑔 {\displaystyle{\displaystyle M\left(\NVar{a},\NVar{g}\right)}}
    xml-id
    C19.S8.SS1.p1.m7acdec
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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.m2acdec
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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.m4adec
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    Q11820
    Defining formula
    n 𝑛 {\displaystyle{\displaystyle n}}
    xml-id
    C19.S1.XMD3.m1ddec
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    Q11808
    Defining formula
    k 𝑘 {\displaystyle{\displaystyle k}}
    xml-id
    C19.S1.XMD5.m1bdec
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    Q11815
    Defining formula
    k ′ superscript 𝑘 ′ {\displaystyle{\displaystyle k^{\prime}}}
    xml-id
    C19.S1.XMD6.m1bdec
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    Q11809
    Defining formula
    α 2 superscript 𝛼 2 {\displaystyle{\displaystyle\alpha^{2}}}
    xml-id
    C19.S1.XMD7.m1dec
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    Q11827
    Defining formula
    Q n subscript 𝑄 𝑛 {\displaystyle{\displaystyle Q_{n}}}
    xml-id
    C19.S8.XMD3.m1dec
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