DLMF:29.8.E9 (Q9405): Difference between revisions

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Latest revision as of 09:07, 3 March 2024

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DLMF:29.8.E9
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    Statements

    DLMF defining formula
    𝐸𝑠 ν 2 ⁢ m + 2 ⁡ ( z 1 , k 2 ) ⁢ d w 2 ⁢ ( z ) / d z | z = K ⁡ - d w 2 ⁢ ( z ) / d z | z = - K ⁡ w 2 ⁢ ( 0 ) = - k 4 k ′ ⁢ sn ⁡ ( z 1 , k ) ⁢ cn ⁡ ( z 1 , k ) ⁢ ∫ - K ⁡ K ⁡ sn ⁡ ( z , k ) ⁢ cn ⁡ ( z , k ) ⁢ d 2 𝖯 ν ⁡ ( y ) d y 2 ⁢ 𝐸𝑠 ν 2 ⁢ m + 2 ⁡ ( z , k 2 ) ⁢ d z . Lame-Es 2 𝑚 2 𝜈 subscript 𝑧 1 superscript 𝑘 2 evaluated-at derivative subscript 𝑤 2 𝑧 𝑧 𝑧 complete-elliptic-integral-first-kind-K 𝑘 evaluated-at derivative subscript 𝑤 2 𝑧 𝑧 𝑧 complete-elliptic-integral-first-kind-K 𝑘 subscript 𝑤 2 0 superscript 𝑘 4 superscript 𝑘 ′ Jacobi-elliptic-sn subscript 𝑧 1 𝑘 Jacobi-elliptic-cn subscript 𝑧 1 𝑘 superscript subscript complete-elliptic-integral-first-kind-K 𝑘 complete-elliptic-integral-first-kind-K 𝑘 Jacobi-elliptic-sn 𝑧 𝑘 Jacobi-elliptic-cn 𝑧 𝑘 derivative shorthand-Ferrers-Legendre-P-first-kind 𝜈 𝑦 𝑦 2 Lame-Es 2 𝑚 2 𝜈 𝑧 superscript 𝑘 2 𝑧 {\displaystyle{\displaystyle\mathit{Es}^{2m+2}_{\nu}\left(z_{1},k^{2}\right)% \frac{\left.\ifrac{\mathrm{d}w_{2}(z)}{\mathrm{d}z}\right|_{z=K}-\left.\ifrac{% \mathrm{d}w_{2}(z)}{\mathrm{d}z}\right|_{z=-K}}{w_{2}(0)}=-\frac{k^{4}}{k^{% \prime}}\operatorname{sn}\left(z_{1},k\right)\operatorname{cn}\left(z_{1},k% \right)\int_{-K}^{K}\operatorname{sn}\left(z,k\right)\operatorname{cn}\left(z,% k\right)\frac{{\mathrm{d}}^{2}\mathsf{P}_{\nu}\left(y\right)}{{\mathrm{d}y}^{2% }}\mathit{Es}^{2m+2}_{\nu}\left(z,k^{2}\right)\mathrm{d}z.}}
    0 references
    Symbols used
    Jacobian elliptic function
    DLMF 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.m2abdec
    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.m2abdec
    0 references
    Q12367
    DLMF defining formula
    𝐸𝑠 ν m ⁡ ( z , k 2 ) Lame-Es 𝑚 𝜈 𝑧 superscript 𝑘 2 {\displaystyle{\displaystyle\mathit{Es}^{\NVar{m}}_{\NVar{\nu}}\left(\NVar{z},% \NVar{k^{2}}\right)}}
    xml-id
    C29.S3.SS4.p1.m6aadec
    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.m1aedec
    0 references
    derivative of $$f$$ with respect to $$x$$
    DLMF 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
    differential of x
    DLMF defining formula
    d x 𝑥 {\displaystyle{\displaystyle\mathrm{d}\NVar{x}}}
    xml-id
    C1.S4.SS4.m1addec
    0 references
    integral
    DLMF defining formula
    ∫ {\displaystyle{\displaystyle\int}}
    xml-id
    C1.S4.SS4.m3addec
    0 references
    Q11566
    DLMF defining formula
    𝖯 ν ⁡ ( x ) = 𝖯 ν 0 ⁡ ( x ) shorthand-Ferrers-Legendre-P-first-kind 𝜈 𝑥 Ferrers-Legendre-P-first-kind 0 𝜈 𝑥 {\displaystyle{\displaystyle\mathsf{P}_{\NVar{\nu}}\left(\NVar{x}\right)=% \mathsf{P}^{0}_{\nu}\left(x\right)}}
    xml-id
    C14.S2.SS2.p2.m2addec
    0 references
    nonnegative integer
    DLMF defining formula
    m 𝑚 {\displaystyle{\displaystyle m}}
    xml-id
    C29.S1.XMD1.m1cdec
    0 references
    real variable
    DLMF defining formula
    y 𝑦 {\displaystyle{\displaystyle y}}
    xml-id
    C29.S1.XMD5.m1ddec
    0 references
    complex variable
    DLMF defining formula
    z 𝑧 {\displaystyle{\displaystyle z}}
    xml-id
    C29.S1.XMD6.m1gdec
    0 references
    real parameter
    DLMF defining formula
    k 𝑘 {\displaystyle{\displaystyle k}}
    xml-id
    C29.S1.XMD8.m1gdec
    0 references
    real parameter
    DLMF defining formula
    ν 𝜈 {\displaystyle{\displaystyle\nu}}
    xml-id
    C29.S1.XMD9.m1ddec
    0 references
    solution
    DLMF defining formula
    w ⁢ ( z ) 𝑤 𝑧 {\displaystyle{\displaystyle w(z)}}
    xml-id
    C29.S8.XMD1.m1fdec
    0 references

    Identifiers

     
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