DLMF:29.8.E7 (Q9403): Difference between revisions

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Property / Symbols used
 
Property / Symbols used: real parameter / rank
 
Normal rank
Property / Symbols used: real parameter / qualifier
 
DLMF defining formula:

k 𝑘 {\displaystyle{\displaystyle k}}

k
Property / Symbols used: real parameter / qualifier
 
xml-id: C29.S1.XMD8.m1edec

Revision as of 02:09, 22 December 2021

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DLMF:29.8.E7
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    Statements

    DLMF defining formula
    𝐸𝑐 ν 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 ν ⁡ ( y ) d y ⁢ 𝐸𝑐 ν 2 ⁢ m + 1 ⁡ ( z , k 2 ) ⁢ d z , Lame-Ec 2 𝑚 1 𝜈 subscript 𝑧 1 superscript 𝑘 2 subscript 𝑤 2 complete-elliptic-integral-first-kind-K 𝑘 subscript 𝑤 2 complete-elliptic-integral-first-kind-K 𝑘 subscript 𝑤 2 0 superscript 𝑘 2 Jacobi-elliptic-sn subscript 𝑧 1 𝑘 superscript subscript complete-elliptic-integral-first-kind-K 𝑘 complete-elliptic-integral-first-kind-K 𝑘 Jacobi-elliptic-sn 𝑧 𝑘 derivative shorthand-Ferrers-Legendre-P-first-kind 𝜈 𝑦 𝑦 Lame-Ec 2 𝑚 1 𝜈 𝑧 superscript 𝑘 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 𝑘 {\displaystyle{\displaystyle\operatorname{sn}\left(\NVar{z},\NVar{k}\right)}}
    xml-id
    C22.S2.E4.m2aadec
    0 references
    Q12365
    DLMF defining formula
    𝐸𝑐 ν m ⁡ ( z , k 2 ) Lame-Ec 𝑚 𝜈 𝑧 superscript 𝑘 2 {\displaystyle{\displaystyle\mathit{Ec}^{\NVar{m}}_{\NVar{\nu}}\left(\NVar{z},% \NVar{k^{2}}\right)}}
    xml-id
    C29.S3.SS4.p1.m5aadec
    0 references
    Legendre’s complete elliptic integral of the first kind
    DLMF defining formula
    K ⁡ ( k ) complete-elliptic-integral-first-kind-K 𝑘 {\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 𝑥 {\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 𝑥 {\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
    0 references
    Q11566
    DLMF defining formula
    𝖯 ν ⁡ ( x ) = 𝖯 ν 0 ⁡ ( x ) shorthand-Ferrers-Legendre-P-first-kind 𝜈 𝑥 Ferrers-Legendre-P-first-kind 0 𝜈 𝑥 {\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 𝑚 {\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
    0 references
    complex variable
    DLMF defining formula
    z 𝑧 {\displaystyle{\displaystyle z}}
    xml-id
    C29.S1.XMD6.m1edec
    0 references
    real parameter
    DLMF defining formula
    k 𝑘 {\displaystyle{\displaystyle k}}
    xml-id
    C29.S1.XMD8.m1edec
    0 references

    Identifiers

     
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