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DLMF:22.16.E33 - MaRDI portal

DLMF:22.16.E33 (Q7846)

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DLMF:22.16.E33
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    Statements

    DLMF defining formula
    Z ⁡ ( x + K | k ) = Z ⁡ ( x | k ) - k 2 ⁢ sn ⁡ ( x , k ) ⁢ cd ⁡ ( x , k ) , Jacobi-Zeta 𝑥 𝐾 𝑘 Jacobi-Zeta 𝑥 𝑘 superscript 𝑘 2 Jacobi-elliptic-sn 𝑥 𝑘 Jacobi-elliptic-cd 𝑥 𝑘 {\displaystyle{\displaystyle\mathrm{Z}\left(x+K|k\right)=\mathrm{Z}\left(x|k% \right)-k^{2}\operatorname{sn}\left(x,k\right)\operatorname{cd}\left(x,k\right% ),}}
    0 references
    Symbols used
    Jacobi’s zeta function
    DLMF defining formula
    Z ⁡ ( x | k ) Jacobi-Zeta 𝑥 𝑘 {\displaystyle{\displaystyle\mathrm{Z}\left(\NVar{x}|\NVar{k}\right)}}
    xml-id
    C22.S16.E32.m2aadec
    0 references
    Jacobian elliptic function
    DLMF defining formula
    cd ⁡ ( z , k ) Jacobi-elliptic-cd 𝑧 𝑘 {\displaystyle{\displaystyle\operatorname{cd}\left(\NVar{z},\NVar{k}\right)}}
    xml-id
    C22.S2.E8.m2addec
    0 references
    Jacobian elliptic function
    DLMF 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.m2aldec
    0 references
    Legendre’s complete elliptic integral of the first kind
    DLMF defining formula
    K ⁡ ( k ) complete-elliptic-integral-first-kind-K 𝑘 {\displaystyle{\displaystyle K\left(\NVar{k}\right)}}
    xml-id
    C19.S2.E8.m1agdec
    0 references
    real
    DLMF defining formula
    x 𝑥 {\displaystyle{\displaystyle x}}
    xml-id
    C22.S1.XMD1.m1afdec
    0 references
    modulus
    DLMF defining formula
    k 𝑘 {\displaystyle{\displaystyle k}}
    xml-id
    C22.S1.XMD4.m1addec
    0 references

    Identifiers

     
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