DLMF:23.6.E29 (Q7272)

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DLMF:23.6.E29
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    Statements

    Defining formula
    ζ ⁡ ( z | 𝕃 3 ) - ζ ⁡ ( z + 2 ⁢ i ⁢ K ′ ⁡ | 𝕃 3 ) - ζ ⁡ ( 2 ⁢ i ⁢ K ′ ⁡ | 𝕃 3 ) = cs ⁡ ( z , k ) . Weierstrass-zeta-on-lattice 𝑧 subscript 𝕃 3 Weierstrass-zeta-on-lattice 𝑧 2 imaginary-unit complementary-complete-elliptic-integral-first-kind-K 𝑘 subscript 𝕃 3 Weierstrass-zeta-on-lattice 2 imaginary-unit complementary-complete-elliptic-integral-first-kind-K 𝑘 subscript 𝕃 3 Jacobi-elliptic-cs 𝑧 𝑘 {\displaystyle{\displaystyle\zeta\left(z|\mathbb{L}_{\mspace{1.0mu }3}\right)-% \zeta\left(z+2\mathrm{i}{K^{\prime}}|\mathbb{L}_{\mspace{1.0mu }3}\right)-% \zeta\left(2\mathrm{i}{K^{\prime}}|\mathbb{L}_{\mspace{1.0mu }3}\right)=% \operatorname{cs}\left(z,k\right).}}
    0 references
    DLMF id
    DLMF:23.6.E29
    0 references
    Symbols used
    Jacobian elliptic function
    Defining formula
    cs ⁡ ( z , k ) Jacobi-elliptic-cs 𝑧 𝑘 {\displaystyle{\displaystyle\operatorname{cs}\left(\NVar{z},\NVar{k}\right)}}
    xml-id
    C22.S2.E9.m3aadec
    0 references
    DLMF:23.2.E5
    Defining formula
    ζ ⁡ ( z ) Weierstrass-zeta-on-lattice 𝑧 𝕃 {\displaystyle{\displaystyle\zeta\left(\NVar{z}\right)}}
    xml-id
    C23.S2.E5.m2acdec
    0 references
    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.m1aadec
    0 references
    imaginary unit
    Defining formula
    i imaginary-unit {\displaystyle{\displaystyle\mathrm{i}}}
    xml-id
    C1.S9.E1.m2abdec
    0 references
    Q12055
    Defining formula
    𝕃 𝕃 {\displaystyle{\displaystyle\mathbb{L}}}
    xml-id
    C23.S1.XMD1.m1zdec
    0 references
    Q12056
    Defining formula
    z 𝑧 {\displaystyle{\displaystyle z}}
    xml-id
    C23.S1.XMD7.m1odec
    0 references
     
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