DLMF:29.8.E9 (Q8713)

From DRMF
Revision as of 01:24, 2 January 2020 by Admin (talk | contribs) (β€ŽChanged an Item: Add constraint)
Jump to navigation Jump to search
No description defined
Language Label Description Also known as
English
DLMF:29.8.E9
No description defined

    Statements

    Defining formula
    𝐸𝑠 Ξ½ 2 ⁒ m + 2 ⁑ ( z 1 , k 2 ) ⁒ d w 2 ⁒ ( z ) / d z | z = K ⁑ - d w 2 ⁒ ( z ) / d z | z = - K ⁑ w 2 ⁒ ( 0 ) = - k 4 k β€² ⁒ sn ⁑ ( z 1 , k ) ⁒ cn ⁑ ( z 1 , k ) ⁒ ∫ - K ⁑ K ⁑ sn ⁑ ( z , k ) ⁒ cn ⁑ ( z , k ) ⁒ d 2 𝖯 Ξ½ ⁑ ( y ) d y 2 ⁒ 𝐸𝑠 Ξ½ 2 ⁒ m + 2 ⁑ ( z , k 2 ) ⁒ d z . Lame-Es 2 π‘š 2 𝜈 subscript 𝑧 1 superscript π‘˜ 2 evaluated-at derivative subscript 𝑀 2 𝑧 𝑧 𝑧 complete-elliptic-integral-first-kind-K π‘˜ evaluated-at derivative subscript 𝑀 2 𝑧 𝑧 𝑧 complete-elliptic-integral-first-kind-K π‘˜ subscript 𝑀 2 0 superscript π‘˜ 4 superscript π‘˜ β€² Jacobi-elliptic-sn subscript 𝑧 1 π‘˜ Jacobi-elliptic-cn subscript 𝑧 1 π‘˜ superscript subscript complete-elliptic-integral-first-kind-K π‘˜ complete-elliptic-integral-first-kind-K π‘˜ Jacobi-elliptic-sn 𝑧 π‘˜ Jacobi-elliptic-cn 𝑧 π‘˜ derivative shorthand-Ferrers-Legendre-P-first-kind 𝜈 𝑦 𝑦 2 Lame-Es 2 π‘š 2 𝜈 𝑧 superscript π‘˜ 2 𝑧 {\displaystyle{\displaystyle\mathit{Es}^{2m+2}_{\nu}\left(z_{1},k^{2}\right)% \frac{\left.\ifrac{\mathrm{d}w_{2}(z)}{\mathrm{d}z}\right|_{z=K}-\left.\ifrac{% \mathrm{d}w_{2}(z)}{\mathrm{d}z}\right|_{z=-K}}{w_{2}(0)}=-\frac{k^{4}}{k^{% \prime}}\operatorname{sn}\left(z_{1},k\right)\operatorname{cn}\left(z_{1},k% \right)\int_{-K}^{K}\operatorname{sn}\left(z,k\right)\operatorname{cn}\left(z,% k\right)\frac{{\mathrm{d}}^{2}\mathsf{P}_{\nu}\left(y\right)}{{\mathrm{d}y}^{2% }}\mathit{Es}^{2m+2}_{\nu}\left(z,k^{2}\right)\mathrm{d}z.}}
    0 references
    DLMF id
    DLMF:29.8.E9
    0 references
    Symbols used
    Jacobian elliptic function
    Defining formula
    cn ⁑ ( z , k ) Jacobi-elliptic-cn 𝑧 π‘˜ {\displaystyle{\displaystyle\operatorname{cn}\left(\NVar{z},\NVar{k}\right)}}
    xml-id
    C22.S2.E5.m2abdec
    0 references
    Jacobian elliptic function
    Defining formula
    sn ⁑ ( z , k ) Jacobi-elliptic-sn 𝑧 π‘˜ {\displaystyle{\displaystyle\operatorname{sn}\left(\NVar{z},\NVar{k}\right)}}
    xml-id
    C22.S2.E4.m2abdec
    0 references
    Q12364
    Defining formula
    𝐸𝑠 Ξ½ m ⁑ ( z , k 2 ) Lame-Es π‘š 𝜈 𝑧 superscript π‘˜ 2 {\displaystyle{\displaystyle\mathit{Es}^{\NVar{m}}_{\NVar{\nu}}\left(\NVar{z},% \NVar{k^{2}}\right)}}
    xml-id
    C29.S3.SS4.p1.m6aadec
    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.m1aedec
    0 references
     
        Retrieved from "https://drmf-beta.wmflabs.org/index.php?title=Item:Q8713&oldid=31150"