DLMF:31.2.E8 (Q8989)

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DLMF:31.2.E8
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    Statements

    Defining formula
    d 2 w d ζ 2 + ( ( 2 ⁢ γ - 1 ) ⁢ cn ⁡ ζ ⁢ dn ⁡ ζ sn ⁡ ζ - ( 2 ⁢ δ - 1 ) ⁢ sn ⁡ ζ ⁢ dn ⁡ ζ cn ⁡ ζ - ( 2 ⁢ ϵ - 1 ) ⁢ k 2 ⁢ sn ⁡ ζ ⁢ cn ⁡ ζ dn ⁡ ζ ) ⁢ d w d ζ + 4 ⁢ k 2 ⁢ ( α ⁢ β ⁢ sn 2 ⁡ ζ - q ) ⁢ w = 0 . derivative 𝑤 𝜁 2 2 𝛾 1 Jacobi-elliptic-cn 𝜁 𝑘 Jacobi-elliptic-dn 𝜁 𝑘 Jacobi-elliptic-sn 𝜁 𝑘 2 𝛿 1 Jacobi-elliptic-sn 𝜁 𝑘 Jacobi-elliptic-dn 𝜁 𝑘 Jacobi-elliptic-cn 𝜁 𝑘 2 italic-ϵ 1 superscript 𝑘 2 Jacobi-elliptic-sn 𝜁 𝑘 Jacobi-elliptic-cn 𝜁 𝑘 Jacobi-elliptic-dn 𝜁 𝑘 derivative 𝑤 𝜁 4 superscript 𝑘 2 𝛼 𝛽 Jacobi-elliptic-sn 2 𝜁 𝑘 𝑞 𝑤 0 {\displaystyle{\displaystyle\frac{{\mathrm{d}}^{2}w}{{\mathrm{d}\zeta}^{2}}+% \left((2\gamma-1)\frac{\operatorname{cn}\zeta\operatorname{dn}\zeta}{% \operatorname{sn}\zeta}-(2\delta-1)\frac{\operatorname{sn}\zeta\operatorname{% dn}\zeta}{\operatorname{cn}\zeta}-(2\epsilon-1)k^{2}\frac{\operatorname{sn}% \zeta\operatorname{cn}\zeta}{\operatorname{dn}\zeta}\right)\frac{\mathrm{d}w}{% \mathrm{d}\zeta}+4k^{2}(\alpha\beta{\operatorname{sn}^{2}}\zeta-q)w=0.}}
    0 references
    DLMF id
    DLMF:31.2.E8
    0 references
    Symbols used
    Jacobian elliptic function
    Defining formula
    cn ⁡ ( z , k ) Jacobi-elliptic-cn 𝑧 𝑘 {\displaystyle{\displaystyle\operatorname{cn}\left(\NVar{z},\NVar{k}\right)}}
    xml-id
    C22.S2.E5.m2adec
    0 references
    Jacobian elliptic function
    Defining formula
    dn ⁡ ( z , k ) Jacobi-elliptic-dn 𝑧 𝑘 {\displaystyle{\displaystyle\operatorname{dn}\left(\NVar{z},\NVar{k}\right)}}
    xml-id
    C22.S2.E6.m2adec
    0 references
    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.m2aadec
    0 references
    derivative of $$f$$ with respect to $$x$$
    Defining formula
    d f d x derivative 𝑓 𝑥 {\displaystyle{\displaystyle\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}}}
    xml-id
    C1.S4.E4.m2acdec
    0 references
    Q12458
    Defining formula
    γ 𝛾 {\displaystyle{\displaystyle\gamma}}
    xml-id
    C31.S1.XMD10.m1ddec
    0 references
    Q12459
    Defining formula
    δ 𝛿 {\displaystyle{\displaystyle\delta}}
    xml-id
    C31.S1.XMD11.m1ddec
    0 references
    Q12460
    Defining formula
    ϵ italic-ϵ {\displaystyle{\displaystyle\epsilon}}
    xml-id
    C31.S1.XMD12.m1ddec
    0 references
    Q12462
    Defining formula
    q 𝑞 {\displaystyle{\displaystyle q}}
    xml-id
    C31.S1.XMD7.m1cdec
    0 references
    Q12463
    Defining formula
    α 𝛼 {\displaystyle{\displaystyle\alpha}}
    xml-id
    C31.S1.XMD8.m1cdec
    0 references
     
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