DLMF:29.15.E45 (Q8790): Difference between revisions

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Property / Symbols used
 
Property / Symbols used: Jacobian elliptic function / rank
 
Normal rank
Property / Symbols used: Jacobian elliptic function / qualifier
 
Defining formula:

sn ⁡ ( z , k ) Jacobi-elliptic-sn 𝑧 𝑘 {\displaystyle{\displaystyle\operatorname{sn}\left(\NVar{z},\NVar{k}\right)}}

\Jacobiellsnk@{\NVar{z}}{\NVar{k}}
Property / Symbols used: Jacobian elliptic function / qualifier
 
xml-id: C22.S2.E4.m2abdec
Property / Symbols used
 
Property / Symbols used: Lamé polynomial / rank
 
Normal rank
Property / Symbols used: Lamé polynomial / qualifier
 
Defining formula:

𝑐𝐸 2 ⁢ n + 1 m ⁡ ( z , k 2 ) Lame-polynomial-cE 𝑚 2 𝑛 1 𝑧 superscript 𝑘 2 {\displaystyle{\displaystyle\mathit{cE}^{\NVar{m}}_{2\NVar{n}+1}\left(\NVar{z}% ,\NVar{k^{2}}\right)}}

\LamepolycE{\NVar{m}}{2\NVar{n}+1}@{\NVar{z}}{\NVar{k^{2}}}
Property / Symbols used: Lamé polynomial / qualifier
 
xml-id: C29.S12.E3.m2aadec
Property / Symbols used
 
Property / Symbols used: Q12357 / rank
 
Normal rank
Property / Symbols used: Q12357 / qualifier
 
Defining formula:

m 𝑚 {\displaystyle{\displaystyle m}}

m
Property / Symbols used: Q12357 / qualifier
 
xml-id: C29.S1.XMD1.m1rdec
Property / Symbols used
 
Property / Symbols used: Q12367 / rank
 
Normal rank
Property / Symbols used: Q12367 / qualifier
 
Defining formula:

n 𝑛 {\displaystyle{\displaystyle n}}

n
Property / Symbols used: Q12367 / qualifier
 
xml-id: C29.S1.XMD2.m1ajdec
Property / Symbols used
 
Property / Symbols used: Q12359 / rank
 
Normal rank
Property / Symbols used: Q12359 / qualifier
 
Defining formula:

p 𝑝 {\displaystyle{\displaystyle p}}

p
Property / Symbols used: Q12359 / qualifier
 
xml-id: C29.S1.XMD3.m1zdec
Property / Symbols used
 
Property / Symbols used: Q12348 / rank
 
Normal rank
Property / Symbols used: Q12348 / qualifier
 
Defining formula:

z 𝑧 {\displaystyle{\displaystyle z}}

z
Property / Symbols used: Q12348 / qualifier
 
xml-id: C29.S1.XMD6.m1jdec
Property / Symbols used
 
Property / Symbols used: Q12350 / rank
 
Normal rank
Property / Symbols used: Q12350 / qualifier
 
Defining formula:

k 𝑘 {\displaystyle{\displaystyle k}}

k
Property / Symbols used: Q12350 / qualifier
 
xml-id: C29.S1.XMD8.m1vdec
Property / Symbols used
 
Property / Symbols used: Q12372 / rank
 
Normal rank
Property / Symbols used: Q12372 / qualifier
 
Defining formula:

B 2 ⁢ p + 1 subscript 𝐵 2 𝑝 1 {\displaystyle{\displaystyle B_{2p+1}}}

B_{2p+1}
Property / Symbols used: Q12372 / qualifier
 
xml-id: C29.S6.XMD9.m1dec

Latest revision as of 01:37, 2 January 2020

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DLMF:29.15.E45
No description defined

    Statements

    Defining formula
    𝑐𝐸 2 ⁢ n + 1 m ⁡ ( z , k 2 ) = cn ⁡ ( z , k ) ⁢ ∑ p = 0 n B 2 ⁢ p + 1 ⁢ U 2 ⁢ p ⁡ ( sn ⁡ ( z , k ) ) , Lame-polynomial-cE 𝑚 2 𝑛 1 𝑧 superscript 𝑘 2 Jacobi-elliptic-cn 𝑧 𝑘 superscript subscript 𝑝 0 𝑛 subscript 𝐵 2 𝑝 1 Chebyshev-polynomial-second-kind-U 2 𝑝 Jacobi-elliptic-sn 𝑧 𝑘 {\displaystyle{\displaystyle\mathit{cE}^{m}_{2n+1}\left(z,k^{2}\right)=% \operatorname{cn}\left(z,k\right)\sum_{p=0}^{n}B_{2p+1}U_{2p}\left(% \operatorname{sn}\left(z,k\right)\right),}}
    0 references
    DLMF id
    DLMF:29.15.E45
    0 references
    Symbols used
    Q11661
    Defining formula
    U n ⁡ ( x ) Chebyshev-polynomial-second-kind-U 𝑛 𝑥 {\displaystyle{\displaystyle U_{\NVar{n}}\left(\NVar{x}\right)}}
    xml-id
    C18.S3.T1.t1.r5.m2adec
    0 references
    Jacobian elliptic function
    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.m2adec
    0 references
    Jacobian elliptic function
    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
    Lamé polynomial
    Defining formula
    𝑐𝐸 2 ⁢ n + 1 m ⁡ ( z , k 2 ) Lame-polynomial-cE 𝑚 2 𝑛 1 𝑧 superscript 𝑘 2 {\displaystyle{\displaystyle\mathit{cE}^{\NVar{m}}_{2\NVar{n}+1}\left(\NVar{z}% ,\NVar{k^{2}}\right)}}
    xml-id
    C29.S12.E3.m2aadec
    0 references
    Q12357
    Defining formula
    m 𝑚 {\displaystyle{\displaystyle m}}
    xml-id
    C29.S1.XMD1.m1rdec
    0 references
    Q12367
    Defining formula
    n 𝑛 {\displaystyle{\displaystyle n}}
    xml-id
    C29.S1.XMD2.m1ajdec
    0 references
    Q12359
    Defining formula
    p 𝑝 {\displaystyle{\displaystyle p}}
    xml-id
    C29.S1.XMD3.m1zdec
    0 references
    Q12348
    Defining formula
    z 𝑧 {\displaystyle{\displaystyle z}}
    xml-id
    C29.S1.XMD6.m1jdec
    0 references
    Q12350
    Defining formula
    k 𝑘 {\displaystyle{\displaystyle k}}
    xml-id
    C29.S1.XMD8.m1vdec
    0 references
    Q12372
    Defining formula
    B 2 ⁢ p + 1 subscript 𝐵 2 𝑝 1 {\displaystyle{\displaystyle B_{2p+1}}}
    xml-id
    C29.S6.XMD9.m1dec
    0 references
     
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