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Statements
lim
k
β
1
-
β‘
πΈπ
Ξ½
m
β‘
(
z
,
k
2
)
πΈπ
Ξ½
m
β‘
(
0
,
k
2
)
=
lim
k
β
1
-
β‘
πΈπ
Ξ½
m
+
1
β‘
(
z
,
k
2
)
πΈπ
Ξ½
m
+
1
β‘
(
0
,
k
2
)
=
1
(
cosh
β‘
z
)
ΞΌ
β’
F
β‘
(
1
2
β’
ΞΌ
-
1
2
β’
Ξ½
,
1
2
β’
ΞΌ
+
1
2
β’
Ξ½
+
1
2
1
2
;
tanh
2
β‘
z
)
,
subscript
β
π
limit-from
1
Lame-Ec
π
π
π§
superscript
π
2
Lame-Ec
π
π
0
superscript
π
2
subscript
β
π
limit-from
1
Lame-Es
π
1
π
π§
superscript
π
2
Lame-Es
π
1
π
0
superscript
π
2
1
superscript
π§
π
Gauss-hypergeometric-F
1
2
π
1
2
π
1
2
π
1
2
π
1
2
1
2
2
π§
{\displaystyle{\displaystyle{\lim_{k\to 1-}\frac{\mathit{Ec}^{m}_{\nu}\left(z,%
k^{2}\right)}{\mathit{Ec}^{m}_{\nu}\left(0,k^{2}\right)}=\lim_{k\to 1-}\frac{%
\mathit{Es}^{m+1}_{\nu}\left(z,k^{2}\right)}{\mathit{Es}^{m+1}_{\nu}\left(0,k^%
{2}\right)}}=\frac{1}{(\cosh z)^{\mu}}F\left({\tfrac{1}{2}\mu-\tfrac{1}{2}\nu,%
\tfrac{1}{2}\mu+\tfrac{1}{2}\nu+\tfrac{1}{2}\atop\tfrac{1}{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)}}
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}}
