DLMF:22.11.E14 (Q7038)

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DLMF:22.11.E14
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    Statements

    Defining formula
    k 2 ⁢ sn 2 ⁡ ( z , k ) = E ′ ⁡ K ′ ⁡ - ( π 2 ⁢ K ′ ) 2 ⁢ ∑ n = - ∞ ∞ ( sech 2 ⁡ ( π 2 ⁢ K ′ ⁡ ⁢ ( z - 2 ⁢ n ⁢ K ⁡ ) ) ) , superscript 𝑘 2 Jacobi-elliptic-sn 2 𝑧 𝑘 complementary-complete-elliptic-integral-second-kind-E 𝑘 complementary-complete-elliptic-integral-first-kind-K 𝑘 superscript 𝜋 2 complementary-complete-elliptic-integral-first-kind-K 2 superscript subscript 𝑛 2 𝜋 2 complementary-complete-elliptic-integral-first-kind-K 𝑘 𝑧 2 𝑛 complete-elliptic-integral-first-kind-K 𝑘 {\displaystyle{\displaystyle k^{2}{\operatorname{sn}^{2}}\left(z,k\right)=% \frac{{E^{\prime}}}{{K^{\prime}}}-\left(\frac{\pi}{2{K^{\prime}}}\right)^{2}% \sum_{n=-\infty}^{\infty}\left({\operatorname{sech}^{2}}\left(\frac{\pi}{2{K^{% \prime}}}(z-2nK)\right)\right),}}
    0 references
    DLMF id
    DLMF:22.11.E14
    0 references
    Symbols used
    Jacobian elliptic function
    Defining formula
    sn ⁡ ( z , k ) Jacobi-elliptic-sn 𝑧 𝑘 {\displaystyle{\displaystyle\operatorname{sn}\left(\NVar{z},\NVar{k}\right)}}
    xml-id
    C22.S2.E4.m2abdec
    0 references
    the ratio of the circumference of a circle to its diameter
    Defining formula
    π {\displaystyle{\displaystyle\pi}}
    xml-id
    C3.S12.E1.m2amdec
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    Q11821
    Defining formula
    K ′ ⁡ ( k ) complementary-complete-elliptic-integral-first-kind-K 𝑘 {\displaystyle{\displaystyle{K^{\prime}}\left(\NVar{k}\right)}}
    xml-id
    C19.S2.E9.m1adec
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    Q11822
    Defining formula
    E ′ ⁡ ( k ) complementary-complete-elliptic-integral-second-kind-E 𝑘 {\displaystyle{\displaystyle{E^{\prime}}\left(\NVar{k}\right)}}
    xml-id
    C19.S2.E9.m2adec
    0 references
    Q11600
    Defining formula
    K ⁡ ( k ) complete-elliptic-integral-first-kind-K 𝑘 {\displaystyle{\displaystyle K\left(\NVar{k}\right)}}
    xml-id
    C19.S2.E8.m1amdec
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    hyperbolic secant function
    Defining formula
    sech ⁡ z 𝑧 {\displaystyle{\displaystyle\operatorname{sech}\NVar{z}}}
    xml-id
    C4.S28.E6.m2adec
    0 references
    Q11985
    Defining formula
    z 𝑧 {\displaystyle{\displaystyle z}}
    xml-id
    C22.S1.XMD3.m1mdec
    0 references
     
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