DLMF:19.8.E9 (Q6217)

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DLMF:19.8.E9
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    Statements

    Defining formula
    Π ⁡ ( α 2 , k ) = π 4 ⁢ M ⁡ ( 1 , k ′ ) ⁢ k 2 k 2 - α 2 ⁢ ∑ n = 0 ∞ Q n , complete-elliptic-integral-third-kind-Pi superscript 𝛼 2 𝑘 𝜋 4 arithmetic-geometric-mean 1 superscript 𝑘 ′ superscript 𝑘 2 superscript 𝑘 2 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)}\frac{k^{2}}{k^{2}-\alpha^{2}}\sum_{n=0}^{\infty}Q_{n},}}
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    DLMF id
    DLMF:19.8.E9
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    constraint
    1 < α 2 < ∞ 1 superscript 𝛼 2 {\displaystyle{\displaystyle 1<\alpha^{2}<\infty}}
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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.m7addec
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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.m2addec
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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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    Q11820
    Defining formula
    n 𝑛 {\displaystyle{\displaystyle n}}
    xml-id
    C19.S1.XMD3.m1fdec
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    Q11808
    Defining formula
    k 𝑘 {\displaystyle{\displaystyle k}}
    xml-id
    C19.S1.XMD5.m1cdec
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    Q11815
    Defining formula
    k ′ superscript 𝑘 ′ {\displaystyle{\displaystyle k^{\prime}}}
    xml-id
    C19.S1.XMD6.m1cdec
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    Q11809
    Defining formula
    α 2 superscript 𝛼 2 {\displaystyle{\displaystyle\alpha^{2}}}
    xml-id
    C19.S1.XMD7.m1adec
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    Q11827
    Defining formula
    Q n subscript 𝑄 𝑛 {\displaystyle{\displaystyle Q_{n}}}
    xml-id
    C19.S8.XMD3.m1bdec
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