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Statements
lim
k
β
1
-
β‘
πΈπ
Ξ½
m
β‘
(
z
,
k
2
)
d
πΈπ
Ξ½
m
β‘
(
z
,
k
2
)
/
d
z
|
z
=
0
=
lim
k
β
1
-
β‘
πΈπ
Ξ½
m
+
1
β‘
(
z
,
k
2
)
d
πΈπ
Ξ½
m
+
1
β‘
(
z
,
k
2
)
/
d
z
|
z
=
0
=
tanh
β‘
z
(
cosh
β‘
z
)
ΞΌ
β’
F
β‘
(
1
2
β’
ΞΌ
-
1
2
β’
Ξ½
+
1
2
,
1
2
β’
ΞΌ
+
1
2
β’
Ξ½
+
1
3
2
;
tanh
2
β‘
z
)
,
subscript
β
π
limit-from
1
Lame-Ec
π
π
π§
superscript
π
2
evaluated-at
derivative
Lame-Ec
π
π
π§
superscript
π
2
π§
π§
0
subscript
β
π
limit-from
1
Lame-Es
π
1
π
π§
superscript
π
2
evaluated-at
derivative
Lame-Es
π
1
π
π§
superscript
π
2
π§
π§
0
π§
superscript
π§
π
Gauss-hypergeometric-F
1
2
π
1
2
π
1
2
1
2
π
1
2
π
1
3
2
2
π§
{\displaystyle{\displaystyle\lim_{k\to 1-}\frac{\mathit{Ec}^{m}_{\nu}\left(z,k%
^{2}\right)}{\left.\ifrac{\mathrm{d}\mathit{Ec}^{m}_{\nu}\left(z,k^{2}\right)}%
{\mathrm{d}z}\right|_{z=0}}=\lim_{k\to 1-}\frac{\mathit{Es}^{m+1}_{\nu}\left(z%
,k^{2}\right)}{\left.\ifrac{\mathrm{d}\mathit{Es}^{m+1}_{\nu}\left(z,k^{2}%
\right)}{\mathrm{d}z}\right|_{z=0}}=\frac{\tanh z}{(\cosh z)^{\mu}}F\left({%
\tfrac{1}{2}\mu-\tfrac{1}{2}\nu+\tfrac{1}{2},\tfrac{1}{2}\mu+\tfrac{1}{2}\nu+1%
\atop\tfrac{3}{2}};{\tanh^{2}}z\right),}}
m
π
{\displaystyle{\displaystyle m}}
F
β‘
(
a
,
b
;
c
;
z
)
Gauss-hypergeometric-F
π
π
π
π§
{\displaystyle{\displaystyle F\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)}}
πΈπ
Ξ½
m
β‘
(
z
,
k
2
)
Lame-Ec
π
π
π§
superscript
π
2
{\displaystyle{\displaystyle\mathit{Ec}^{\NVar{m}}_{\NVar{\nu}}\left(\NVar{z},%
\NVar{k^{2}}\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)}}
d
f
d
x
derivative
π
π₯
{\displaystyle{\displaystyle\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}}}
cosh
β‘
z
π§
{\displaystyle{\displaystyle\cosh\NVar{z}}}
tanh
β‘
z
π§
{\displaystyle{\displaystyle\tanh\NVar{z}}}
m
π
{\displaystyle{\displaystyle m}}
z
π§
{\displaystyle{\displaystyle z}}
k
π
{\displaystyle{\displaystyle k}}
Ξ½
π
{\displaystyle{\displaystyle\nu}}
ΞΌ
π
{\displaystyle{\displaystyle\mu}}
