DLMF:29.15.E49 (Q8794): 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:

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

\Jacobielldnk@{\NVar{z}}{\NVar{k}}
Property / Symbols used: Jacobian elliptic function / qualifier
 
xml-id: C22.S2.E6.m2afdec
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.m2afdec
Property / Symbols used
 
Property / Symbols used: Lamé polynomial / rank
 
Normal rank
Property / Symbols used: Lamé polynomial / qualifier
 
Defining formula:

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

\LamepolycdE{\NVar{m}}{2\NVar{n}+2}@{\NVar{z}}{\NVar{k^{2}}}
Property / Symbols used: Lamé polynomial / qualifier
 
xml-id: C29.S12.E7.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.m1vdec
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.m1andec
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.m1addec
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.m1ndec
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.m1zdec
Property / Symbols used
 
Property / Symbols used: Q12375 / rank
 
Normal rank
Property / Symbols used: Q12375 / qualifier
 
Defining formula:

D 2 ⁢ p subscript 𝐷 2 𝑝 {\displaystyle{\displaystyle D_{2p}}}

D_{2p}
Property / Symbols used: Q12375 / qualifier
 
xml-id: C29.S6.XMD13.m1hdec

Latest revision as of 01:38, 2 January 2020

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DLMF:29.15.E49
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    Statements

    Defining formula
    𝑐𝑑𝐸 2 ⁢ n + 2 m ⁡ ( z , k 2 ) = cn ⁡ ( z , k ) ⁢ dn ⁡ ( z , k ) ⁢ ∑ p = 0 n D 2 ⁢ p + 1 ⁢ U 2 ⁢ p ⁡ ( sn ⁡ ( z , k ) ) , Lame-polynomial-cdE 𝑚 2 𝑛 2 𝑧 superscript 𝑘 2 Jacobi-elliptic-cn 𝑧 𝑘 Jacobi-elliptic-dn 𝑧 𝑘 superscript subscript 𝑝 0 𝑛 subscript 𝐷 2 𝑝 1 Chebyshev-polynomial-second-kind-U 2 𝑝 Jacobi-elliptic-sn 𝑧 𝑘 {\displaystyle{\displaystyle\mathit{cdE}^{m}_{2n+2}\left(z,k^{2}\right)=% \operatorname{cn}\left(z,k\right)\operatorname{dn}\left(z,k\right)\sum_{p=0}^{% n}D_{2p+1}U_{2p}\left(\operatorname{sn}\left(z,k\right)\right),}}
    0 references
    DLMF id
    DLMF:29.15.E49
    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.m2abdec
    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.m2abdec
    0 references
    Jacobian elliptic function
    Defining formula
    dn ⁡ ( z , k ) Jacobi-elliptic-dn 𝑧 𝑘 {\displaystyle{\displaystyle\operatorname{dn}\left(\NVar{z},\NVar{k}\right)}}
    xml-id
    C22.S2.E6.m2afdec
    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.m2afdec
    0 references
    Lamé polynomial
    Defining formula
    𝑐𝑑𝐸 2 ⁢ n + 2 m ⁡ ( z , k 2 ) Lame-polynomial-cdE 𝑚 2 𝑛 2 𝑧 superscript 𝑘 2 {\displaystyle{\displaystyle\mathit{cdE}^{\NVar{m}}_{2\NVar{n}+2}\left(\NVar{z% },\NVar{k^{2}}\right)}}
    xml-id
    C29.S12.E7.m2aadec
    0 references
    Q12357
    Defining formula
    m 𝑚 {\displaystyle{\displaystyle m}}
    xml-id
    C29.S1.XMD1.m1vdec
    0 references
    Q12367
    Defining formula
    n 𝑛 {\displaystyle{\displaystyle n}}
    xml-id
    C29.S1.XMD2.m1andec
    0 references
    Q12359
    Defining formula
    p 𝑝 {\displaystyle{\displaystyle p}}
    xml-id
    C29.S1.XMD3.m1addec
    0 references
    Q12348
    Defining formula
    z 𝑧 {\displaystyle{\displaystyle z}}
    xml-id
    C29.S1.XMD6.m1ndec
    0 references
    Q12350
    Defining formula
    k 𝑘 {\displaystyle{\displaystyle k}}
    xml-id
    C29.S1.XMD8.m1zdec
    0 references
    Q12375
    Defining formula
    D 2 ⁢ p subscript 𝐷 2 𝑝 {\displaystyle{\displaystyle D_{2p}}}
    xml-id
    C29.S6.XMD13.m1hdec
    0 references
     
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