DLMF:23.6.E26 (Q7269)

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DLMF:23.6.E26
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    Statements

    Defining formula
    ℘ ⁡ ( z + ω 3 ) - e 3 ⁡ = ( K ⁡ ⁢ k ω 1 ) 2 ⁢ sn 2 ⁡ ( K ⁡ ⁢ z ω 1 , k ) . Weierstrass-P-on-lattice 𝑧 subscript 𝜔 3 𝕃 Weierstrass-e-on-lattice 3 𝕃 superscript complete-elliptic-integral-first-kind-K 𝑘 𝑘 subscript 𝜔 1 2 Jacobi-elliptic-sn 2 complete-elliptic-integral-first-kind-K 𝑘 𝑧 subscript 𝜔 1 𝑘 {\displaystyle{\displaystyle\wp\left(z+\omega_{3}\right)-e_{3}=\left(\frac{Kk}% {\omega_{1}}\right)^{2}{\operatorname{sn}^{2}}\left(\frac{K\!z}{\omega_{1}},k% \right).}}
    0 references
    DLMF id
    DLMF:23.6.E26
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    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.m2adec
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    DLMF:23.2.E4
    Defining formula
    ℘ ⁡ ( z ) Weierstrass-P-on-lattice 𝑧 𝕃 {\displaystyle{\displaystyle\wp\left(\NVar{z}\right)}}
    xml-id
    C23.S2.E4.m2aidec
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    Q11869
    Defining formula
    e j ⁡ Weierstrass-e-on-lattice 𝑗 𝕃 {\displaystyle{\displaystyle e_{\NVar{j}}}}
    xml-id
    C23.S3.SS1.m1apdec
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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.m1aidec
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    Q12055
    Defining formula
    𝕃 𝕃 {\displaystyle{\displaystyle\mathbb{L}}}
    xml-id
    C23.S1.XMD1.m1wdec
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    Q12056
    Defining formula
    z 𝑧 {\displaystyle{\displaystyle z}}
    xml-id
    C23.S1.XMD7.m1ldec
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    Q12049
    Defining formula
    ω 1 subscript 𝜔 1 {\displaystyle{\displaystyle\omega_{1}}}
    xml-id
    C23.S1.XMD8.m1xdec
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    Q12062
    Defining formula
    k 𝑘 {\displaystyle{\displaystyle k}}
    xml-id
    C23.S6.XMD3.m1jdec
    0 references
     
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