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Statements
πΈπ
Ξ½
2
β’
m
+
2
β‘
(
z
1
,
k
2
)
β’
d
w
2
β’
(
z
)
/
d
z
|
z
=
K
β‘
-
d
w
2
β’
(
z
)
/
d
z
|
z
=
-
K
β‘
w
2
β’
(
0
)
=
-
k
4
k
β²
β’
sn
β‘
(
z
1
,
k
)
β’
cn
β‘
(
z
1
,
k
)
β’
β«
-
K
β‘
K
β‘
sn
β‘
(
z
,
k
)
β’
cn
β‘
(
z
,
k
)
β’
d
2
π―
Ξ½
β‘
(
y
)
d
y
2
β’
πΈπ
Ξ½
2
β’
m
+
2
β‘
(
z
,
k
2
)
β’
d
z
.
Lame-Es
2
π
2
π
subscript
π§
1
superscript
π
2
evaluated-at
derivative
subscript
π€
2
π§
π§
π§
complete-elliptic-integral-first-kind-K
π
evaluated-at
derivative
subscript
π€
2
π§
π§
π§
complete-elliptic-integral-first-kind-K
π
subscript
π€
2
0
superscript
π
4
superscript
π
β²
Jacobi-elliptic-sn
subscript
π§
1
π
Jacobi-elliptic-cn
subscript
π§
1
π
superscript
subscript
complete-elliptic-integral-first-kind-K
π
complete-elliptic-integral-first-kind-K
π
Jacobi-elliptic-sn
π§
π
Jacobi-elliptic-cn
π§
π
derivative
shorthand-Ferrers-Legendre-P-first-kind
π
π¦
π¦
2
Lame-Es
2
π
2
π
π§
superscript
π
2
π§
{\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.}}
cn
β‘
(
z
,
k
)
Jacobi-elliptic-cn
π§
π
{\displaystyle{\displaystyle\operatorname{cn}\left(\NVar{z},\NVar{k}\right)}}
sn
β‘
(
z
,
k
)
Jacobi-elliptic-sn
π§
π
{\displaystyle{\displaystyle\operatorname{sn}\left(\NVar{z},\NVar{k}\right)}}
πΈπ
Ξ½
m
β‘
(
z
,
k
2
)
Lame-Es
π
π
π§
superscript
π
2
{\displaystyle{\displaystyle\mathit{Es}^{\NVar{m}}_{\NVar{\nu}}\left(\NVar{z},%
\NVar{k^{2}}\right)}}
K
β‘
(
k
)
complete-elliptic-integral-first-kind-K
π
{\displaystyle{\displaystyle K\left(\NVar{k}\right)}}
d
f
d
x
derivative
π
π₯
{\displaystyle{\displaystyle\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}}}
d
x
π₯
{\displaystyle{\displaystyle\mathrm{d}\NVar{x}}}
β«
{\displaystyle{\displaystyle\int}}
π―
Ξ½
β‘
(
x
)
=
π―
Ξ½
0
β‘
(
x
)
shorthand-Ferrers-Legendre-P-first-kind
π
π₯
Ferrers-Legendre-P-first-kind
0
π
π₯
{\displaystyle{\displaystyle\mathsf{P}_{\NVar{\nu}}\left(\NVar{x}\right)=%
\mathsf{P}^{0}_{\nu}\left(x\right)}}
m
π
{\displaystyle{\displaystyle m}}
y
π¦
{\displaystyle{\displaystyle y}}
z
π§
{\displaystyle{\displaystyle z}}
k
π
{\displaystyle{\displaystyle k}}
Ξ½
π
{\displaystyle{\displaystyle\nu}}
w
β’
(
z
)
π€
π§
{\displaystyle{\displaystyle w(z)}}
