DLMF:29.8.E7 (Q9403)

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DLMF:29.8.E7
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    Statements

    DLMF defining formula
    𝐸𝑐 Ξ½ 2 ⁒ m + 1 ⁑ ( z 1 , k 2 ) ⁒ w 2 ⁒ ( K ⁑ ) + w 2 ⁒ ( - K ⁑ ) w 2 ⁒ ( 0 ) = - k 2 ⁒ sn ⁑ ( z 1 , k ) ⁒ ∫ - K ⁑ K ⁑ sn ⁑ ( z , k ) ⁒ d 𝖯 Ξ½ ⁑ ( y ) d y ⁒ 𝐸𝑐 Ξ½ 2 ⁒ m + 1 ⁑ ( z , k 2 ) ⁒ d z , Lame-Ec 2 π‘š 1 𝜈 subscript 𝑧 1 superscript π‘˜ 2 subscript 𝑀 2 complete-elliptic-integral-first-kind-K π‘˜ subscript 𝑀 2 complete-elliptic-integral-first-kind-K π‘˜ subscript 𝑀 2 0 superscript π‘˜ 2 Jacobi-elliptic-sn 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‘˜ derivative shorthand-Ferrers-Legendre-P-first-kind 𝜈 𝑦 𝑦 Lame-Ec 2 π‘š 1 𝜈 𝑧 superscript π‘˜ 2 𝑧 {\displaystyle{\displaystyle\mathit{Ec}^{2m+1}_{\nu}\left(z_{1},k^{2}\right)% \frac{w_{2}(K)+w_{2}(-K)}{w_{2}(0)}=-k^{2}\operatorname{sn}\left(z_{1},k\right% )\int_{-K}^{K}\operatorname{sn}\left(z,k\right)\frac{\mathrm{d}\mathsf{P}_{\nu% }\left(y\right)}{\mathrm{d}y}\mathit{Ec}^{2m+1}_{\nu}\left(z,k^{2}\right)% \mathrm{d}z,}}
    0 references
    Symbols used
    Jacobian elliptic function
    DLMF 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.m2aadec
    0 references
    Q12365
    DLMF defining formula
    𝐸𝑐 Ξ½ m ⁑ ( z , k 2 ) Lame-Ec π‘š 𝜈 𝑧 superscript π‘˜ 2 {\displaystyle{\displaystyle\mathit{Ec}^{\NVar{m}}_{\NVar{\nu}}\left(\NVar{z},% \NVar{k^{2}}\right)}}
    xml-id
    C29.S3.SS4.p1.m5aadec
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    Legendre’s complete elliptic integral of the first kind
    DLMF defining formula
    K ⁑ ( k ) complete-elliptic-integral-first-kind-K π‘˜ {\displaystyle{\displaystyle K\left(\NVar{k}\right)}}
    xml-id
    C19.S2.E8.m1acdec
    0 references
    derivative of $$f$$ with respect to $$x$$
    DLMF 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.m2aadec
    0 references
    differential of x
    DLMF defining formula
    d x π‘₯ {\displaystyle{\displaystyle\mathrm{d}\NVar{x}}}
    xml-id
    C1.S4.SS4.m1abdec
    0 references
    integral
    DLMF defining formula
    ∫ {\displaystyle{\displaystyle\int}}
    xml-id
    C1.S4.SS4.m3abdec
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    Q11566
    DLMF 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.m2abdec
    0 references
    nonnegative integer
    DLMF defining formula
    m π‘š {\displaystyle{\displaystyle m}}
    xml-id
    C29.S1.XMD1.m1adec
    0 references
    real variable
    DLMF defining formula
    y 𝑦 {\displaystyle{\displaystyle y}}
    xml-id
    C29.S1.XMD5.m1bdec
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    complex variable
    DLMF defining formula
    z 𝑧 {\displaystyle{\displaystyle z}}
    xml-id
    C29.S1.XMD6.m1edec
    0 references
    real parameter
    DLMF defining formula
    k π‘˜ {\displaystyle{\displaystyle k}}
    xml-id
    C29.S1.XMD8.m1edec
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    real parameter
    DLMF defining formula
    ν 𝜈 {\displaystyle{\displaystyle\nu}}
    xml-id
    C29.S1.XMD9.m1bdec
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    solution
    DLMF defining formula
    w ⁒ ( z ) 𝑀 𝑧 {\displaystyle{\displaystyle w(z)}}
    xml-id
    C29.S8.XMD1.m1ddec
    0 references
     
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