DLMF:29.8.E9 (Q8713)

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DLMF:29.8.E9
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    Statements

    Defining formula
    𝐸𝑠 Ξ½ 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 β€² ⁒ sn ⁑ ( z 1 , k ) ⁒ cn ⁑ ( z 1 , k ) ⁒ ∫ - K ⁑ K ⁑ sn ⁑ ( z , k ) ⁒ cn ⁑ ( z , k ) ⁒ d 2 𝖯 Ξ½ ⁑ ( y ) d y 2 ⁒ 𝐸𝑠 Ξ½ 2 ⁒ m + 2 ⁑ ( z , k 2 ) ⁒ d z . Lame-Es 2 π‘š 2 𝜈 subscript 𝑧 1 superscript π‘˜ 2 evaluated-at derivative subscript 𝑀 2 𝑧 𝑧 𝑧 complete-elliptic-integral-first-kind-K π‘˜ evaluated-at derivative subscript 𝑀 2 𝑧 𝑧 𝑧 complete-elliptic-integral-first-kind-K π‘˜ subscript 𝑀 2 0 superscript π‘˜ 4 superscript π‘˜ β€² Jacobi-elliptic-sn subscript 𝑧 1 π‘˜ Jacobi-elliptic-cn subscript 𝑧 1 π‘˜ superscript subscript complete-elliptic-integral-first-kind-K π‘˜ complete-elliptic-integral-first-kind-K π‘˜ Jacobi-elliptic-sn 𝑧 π‘˜ Jacobi-elliptic-cn 𝑧 π‘˜ derivative shorthand-Ferrers-Legendre-P-first-kind 𝜈 𝑦 𝑦 2 Lame-Es 2 π‘š 2 𝜈 𝑧 superscript π‘˜ 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.}}
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    DLMF id
    DLMF:29.8.E9
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    Symbols used
    Jacobian elliptic function
    Defining formula
    cn ⁑ ( z , k ) Jacobi-elliptic-cn 𝑧 π‘˜ {\displaystyle{\displaystyle\operatorname{cn}\left(\NVar{z},\NVar{k}\right)}}
    xml-id
    C22.S2.E5.m2abdec
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    Jacobian elliptic function
    Defining formula
    sn ⁑ ( z , k ) Jacobi-elliptic-sn 𝑧 π‘˜ {\displaystyle{\displaystyle\operatorname{sn}\left(\NVar{z},\NVar{k}\right)}}
    xml-id
    C22.S2.E4.m2abdec
    0 references
    Q12364
    Defining formula
    𝐸𝑠 Ξ½ m ⁑ ( z , k 2 ) Lame-Es π‘š 𝜈 𝑧 superscript π‘˜ 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
    Q11600
    Defining formula
    K ⁑ ( k ) complete-elliptic-integral-first-kind-K π‘˜ {\displaystyle{\displaystyle K\left(\NVar{k}\right)}}
    xml-id
    C19.S2.E8.m1aedec
    0 references
    derivative of $$f$$ with respect to $$x$$
    Defining formula
    d f d x derivative 𝑓 π‘₯ {\displaystyle{\displaystyle\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}}}
    xml-id
    C1.S4.E4.m2acdec
    0 references
    Q10770
    Defining formula
    d x π‘₯ {\displaystyle{\displaystyle\mathrm{d}\NVar{x}}}
    xml-id
    C1.S4.SS4.m1addec
    0 references
    Q10771
    Defining formula
    ∫ {\displaystyle{\displaystyle\int}}
    xml-id
    C1.S4.SS4.m3addec
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    Q11595
    Defining formula
    𝖯 Ξ½ ⁑ ( x ) = 𝖯 Ξ½ 0 ⁑ ( x ) shorthand-Ferrers-Legendre-P-first-kind 𝜈 π‘₯ Ferrers-Legendre-P-first-kind 0 𝜈 π‘₯ {\displaystyle{\displaystyle\mathsf{P}_{\NVar{\nu}}\left(\NVar{x}\right)=% \mathsf{P}^{0}_{\nu}\left(x\right)}}
    xml-id
    C14.S2.SS2.p2.m2addec
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    Q12357
    Defining formula
    m π‘š {\displaystyle{\displaystyle m}}
    xml-id
    C29.S1.XMD1.m1cdec
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    Q12382
    Defining formula
    y 𝑦 {\displaystyle{\displaystyle y}}
    xml-id
    C29.S1.XMD5.m1ddec
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    Q12348
    Defining formula
    z 𝑧 {\displaystyle{\displaystyle z}}
    xml-id
    C29.S1.XMD6.m1gdec
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    Q12350
    Defining formula
    k π‘˜ {\displaystyle{\displaystyle k}}
    xml-id
    C29.S1.XMD8.m1gdec
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    Q12351
    Defining formula
    ν 𝜈 {\displaystyle{\displaystyle\nu}}
    xml-id
    C29.S1.XMD9.m1ddec
    0 references
     
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