DLMF:29.8.E5 (Q8709)

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DLMF:29.8.E5
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    Statements

    Defining formula
    𝐸𝑐 Ξ½ 2 ⁒ m ⁑ ( z 1 , k 2 ) ⁒ w 2 ⁒ ( K ⁑ ) - w 2 ⁒ ( - K ⁑ ) d w 2 ⁒ ( z ) / d z | z = 0 = ∫ - K ⁑ K ⁑ 𝖯 Ξ½ ⁑ ( y ) ⁒ 𝐸𝑐 Ξ½ 2 ⁒ m ⁑ ( z , k 2 ) ⁒ d z , Lame-Ec 2 π‘š 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‘˜ evaluated-at derivative subscript 𝑀 2 𝑧 𝑧 𝑧 0 superscript subscript complete-elliptic-integral-first-kind-K π‘˜ complete-elliptic-integral-first-kind-K π‘˜ shorthand-Ferrers-Legendre-P-first-kind 𝜈 𝑦 Lame-Ec 2 π‘š 𝜈 𝑧 superscript π‘˜ 2 𝑧 {\displaystyle{\displaystyle\mathit{Ec}^{2m}_{\nu}\left(z_{1},k^{2}\right)% \frac{w_{2}(K)-w_{2}(-K)}{\left.\ifrac{\mathrm{d}w_{2}(z)}{\mathrm{d}z}\right|% _{z=0}}=\int_{-K}^{K}\mathsf{P}_{\nu}\left(y\right)\mathit{Ec}^{2m}_{\nu}\left% (z,k^{2}\right)\mathrm{d}z,}}
    0 references
    DLMF id
    DLMF:29.8.E5
    0 references
    Symbols used
    Q12362
    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.m5adec
    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.m1abdec
    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.m2adec
    0 references
    Q10770
    Defining formula
    d x π‘₯ {\displaystyle{\displaystyle\mathrm{d}\NVar{x}}}
    xml-id
    C1.S4.SS4.m1aadec
    0 references
    Q10771
    Defining formula
    ∫ {\displaystyle{\displaystyle\int}}
    xml-id
    C1.S4.SS4.m3aadec
    0 references
    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.m2aadec
    0 references
    Q12357
    Defining formula
    m π‘š {\displaystyle{\displaystyle m}}
    xml-id
    C29.S1.XMD1.m1dec
    0 references
    Q12382
    Defining formula
    y 𝑦 {\displaystyle{\displaystyle y}}
    xml-id
    C29.S1.XMD5.m1dec
    0 references
    Q12348
    Defining formula
    z 𝑧 {\displaystyle{\displaystyle z}}
    xml-id
    C29.S1.XMD6.m1cdec
    0 references
    Q12350
    Defining formula
    k π‘˜ {\displaystyle{\displaystyle k}}
    xml-id
    C29.S1.XMD8.m1cdec
    0 references
     
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