DLMF:29.8.E9
type: Item
Digital Library of Mathematical Functions ID: 29.8.E9
DLMF defining formula: <math xmlns="http://www.w3.org/1998/Math/MathML" id="p1.1.m1.1" class="ltx_Math" alttext="{\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.}}" display="inline"> 𝐸𝑠 ν 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 ′ ⁢ sn ⁡ ( z 1 , k ) ⁢ cn ⁡ ( z 1 , k ) ⁢ ∫ - K ⁡ K ⁡ sn ⁡ ( z , k ) ⁢ cn ⁡ ( z , k ) ⁢ <mpadded lspace="-1.7pt" width="-4.5pt" id="p1.1.m1.1.13.2.2.1.1" xref="p1.1.m1.1.13.1.cmml"> d 2 </mpadded> 𝖯 ν ⁡ ( y ) d y 2 ⁢ 𝐸𝑠 ν 2 ⁢ m + 2 ⁡ ( z , k 2 ) ⁢ d z . <annotation-xml encoding="MathML-Content" id="p1.1.m1.1b"> <apply id="p1.1.m1.1.17.2.cmml" xref="p1.1.m1.1.17"> <eq id="p1.1.m1.1.3.cmml" xref="p1.1.m1.1.3"/> <apply id="p1.1.m1.1.17.2.1.cmml" xref="p1.1.m1.1.17.2.1"> <times id="p1.1.m1.1.17.2.1.1.cmml" xref="p1.1.m1.1.17.2.1.1"/> <apply id="p1.1.m1.1.1.1.cmml" xref="p1.1.m1.1.1.9.2"> <apply id="p1.1.m1.1.1.2.1.1.cmml" xref="p1.1.m1.1.1.2.1.6"> <csymbol cd="dlmf" id="p1.1.m1.1.1.2.1.1.1.cmml" xref="p1.1.m1.1.1.2.1.3">Lame-Es</csymbol> <apply id="p1.1.m1.1.1.2.1.4.1.cmml" xref="p1.1.m1.1.1.2.1.4.1"> <plus id="p1.1.m1.1.1.2.1.4.1.1.3.cmml" xref="p1.1.m1.1.1.2.1.4.1.1.3"/> <apply id="p1.1.m1.1.1.2.1.4.1.1.5.cmml" xref="p1.1.m1.1.1.2.1.4.1.1.5"> <times id="p1.1.m1.1.1.2.1.4.1.1.5.1.cmml" xref="p1.1.m1.1.1.2.1.4.1.1.5.1"/> <cn type="integer" id="p1.1.m1.1.1.2.1.4.1.1.1.cmml" xref="p1.1.m1.1.1.2.1.4.1.1.1">2</cn> <ci id="p1.1.m1.1.1.2.1.4.1.1.2.cmml" xref="p1.1.m1.1.1.2.1.4.1.1.2">𝑚</ci> </apply> <cn type="integer" id="p1.1.m1…
Symbols used: Jacobian elliptic function
Symbols used: Q12367
Symbols used: Legendre’s complete elliptic integral of the first kind
Symbols used: derivative of $$f$$ with respect to $$x$$
Symbols used: differential of x
Symbols used: integral
Symbols used: Q11566
Symbols used: nonnegative integer
Symbols used: real variable
Symbols used: complex variable
Symbols used: real parameter
Symbols used: solution
instance of: Digital Library of Mathematical Functions Formula
MaRDI profile type: MaRDI formula profile