DLMF:29.15.E28 (Q8773)

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DLMF:29.15.E28
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    Statements

    Defining formula
    𝑠𝑑𝐸 2 ⁒ n + 2 m ⁑ ( z , k 2 ) = dn ⁑ ( z , k ) ⁒ βˆ‘ p = 0 n C 2 ⁒ p + 1 ⁒ cos ⁑ ( ( 2 ⁒ p + 1 ) ⁒ Ο• ) . Lame-polynomial-sdE π‘š 2 𝑛 2 𝑧 superscript π‘˜ 2 Jacobi-elliptic-dn 𝑧 π‘˜ superscript subscript 𝑝 0 𝑛 subscript 𝐢 2 𝑝 1 2 𝑝 1 italic-Ο• {\displaystyle{\displaystyle\mathit{sdE}^{m}_{2n+2}\left(z,k^{2}\right)=% \operatorname{dn}\left(z,k\right)\sum_{p=0}^{n}C_{2p+1}\cos\left((2p+1)\phi% \right).}}
    0 references
    DLMF id
    DLMF:29.15.E28
    0 references
    Symbols used
    Jacobian elliptic function
    Defining formula
    dn ⁑ ( z , k ) Jacobi-elliptic-dn 𝑧 π‘˜ {\displaystyle{\displaystyle\operatorname{dn}\left(\NVar{z},\NVar{k}\right)}}
    xml-id
    C22.S2.E6.m2aadec
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    LamΓ© polynomial
    Defining formula
    𝑠𝑑𝐸 2 ⁒ n + 2 m ⁑ ( z , k 2 ) Lame-polynomial-sdE π‘š 2 𝑛 2 𝑧 superscript π‘˜ 2 {\displaystyle{\displaystyle\mathit{sdE}^{\NVar{m}}_{2\NVar{n}+2}\left(\NVar{z% },\NVar{k^{2}}\right)}}
    xml-id
    C29.S12.E6.m2adec
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    cosine function
    Defining formula
    cos ⁑ z 𝑧 {\displaystyle{\displaystyle\cos\NVar{z}}}
    xml-id
    C4.S14.E2.m2acdec
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    Q12357
    Defining formula
    m π‘š {\displaystyle{\displaystyle m}}
    xml-id
    C29.S1.XMD1.m1jdec
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    Q12367
    Defining formula
    n 𝑛 {\displaystyle{\displaystyle n}}
    xml-id
    C29.S1.XMD2.m1vdec
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    Q12359
    Defining formula
    p 𝑝 {\displaystyle{\displaystyle p}}
    xml-id
    C29.S1.XMD3.m1odec
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    Q12348
    Defining formula
    z 𝑧 {\displaystyle{\displaystyle z}}
    xml-id
    C29.S1.XMD6.m1edec
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    Q12350
    Defining formula
    k π‘˜ {\displaystyle{\displaystyle k}}
    xml-id
    C29.S1.XMD8.m1kdec
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    change of variable (locally)
    Defining formula
    Ο• italic-Ο• {\displaystyle{\displaystyle\phi}}
    xml-id
    C29.S2.XMD3.m1edec
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    Q12371
    Defining formula
    C 2 ⁒ p + 1 subscript 𝐢 2 𝑝 1 {\displaystyle{\displaystyle C_{2p+1}}}
    xml-id
    C29.S6.XMD7.m1ddec
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