DLMF:19.5.E9 (Q6138)

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DLMF:19.5.E9
No description defined

    Statements

    Defining formula
    E ⁡ ( k ) = K ⁡ ( k ) + 2 ⁢ π 2 K ⁡ ( k ) ⁢ ∑ n = 1 ∞ ( - 1 ) n ⁢ n 2 ⁢ q n 2 1 + 2 ⁢ ∑ n = 1 ∞ ( - 1 ) n ⁢ q n 2 , complete-elliptic-integral-second-kind-E 𝑘 complete-elliptic-integral-first-kind-K 𝑘 2 superscript 𝜋 2 complete-elliptic-integral-first-kind-K 𝑘 superscript subscript 𝑛 1 superscript 1 𝑛 superscript 𝑛 2 superscript 𝑞 superscript 𝑛 2 1 2 superscript subscript 𝑛 1 superscript 1 𝑛 superscript 𝑞 superscript 𝑛 2 {\displaystyle{\displaystyle E\left(k\right)=K\left(k\right)+\frac{2\pi^{2}}{K% \left(k\right)}\,\frac{\sum_{n=1}^{\infty}(-1)^{n}n^{2}q^{n^{2}}}{1+2\sum_{n=1% }^{\infty}(-1)^{n}q^{n^{2}}},}}
    0 references
    DLMF id
    DLMF:19.5.E9
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    constraint
    | q | < 1 𝑞 1 {\displaystyle{\displaystyle|q|<1}}
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    Symbols used
    the ratio of the circumference of a circle to its diameter
    Defining formula
    π {\displaystyle{\displaystyle\pi}}
    xml-id
    C3.S12.E1.m2afdec
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    Q11600
    Defining formula
    K ⁡ ( k ) complete-elliptic-integral-first-kind-K 𝑘 {\displaystyle{\displaystyle K\left(\NVar{k}\right)}}
    xml-id
    C19.S2.E8.m1acdec
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    Q11601
    Defining formula
    E ⁡ ( k ) complete-elliptic-integral-second-kind-E 𝑘 {\displaystyle{\displaystyle E\left(\NVar{k}\right)}}
    xml-id
    C19.S2.E8.m2aadec
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    nome
    Defining formula
    q 𝑞 {\displaystyle{\displaystyle q}}
    xml-id
    C22.S2.E1.m2acdec
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    Q11820
    Defining formula
    n 𝑛 {\displaystyle{\displaystyle n}}
    xml-id
    C19.S1.XMD3.m1bdec
    0 references
    Q11808
    Defining formula
    k 𝑘 {\displaystyle{\displaystyle k}}
    xml-id
    C19.S1.XMD5.m1gdec
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