Die Elliptische Lambda-Funktion , auch Modulare Lambda-Funktion genannt, ist eine holomorphe modulare Funktion auf der oberen Halbebene der komplexen Zahlen. Sie ist eine Kongruenzuntergruppe vom Typ Γ(2). Sie wird als Hauptmodul für die modulare Kurve X (2) beschrieben.
Komplexe Ebenendarstellung der elliptischen Lambdafunktion Die Elliptische Lambda-Funktion ist auf folgende Weise definiert:
Sei
H
{\displaystyle \mathbb {H} }
λ
:
H
→
C
{\displaystyle \lambda \colon \mathbb {H} \to \mathbb {C} }
Ausdruck über die Jacobi-Thetafunktion :
λ
(
τ
)
=
ϑ
10
4
[
exp
(
i
π
τ
)
]
ϑ
00
4
[
exp
(
i
π
τ
)
]
{\displaystyle \lambda (\tau )={\frac {\vartheta _{10}^{4}[\exp(i\pi \tau )]}{\vartheta _{00}^{4}[\exp(i\pi \tau )]}}}
Dabei gilt:
ϑ
10
[
exp
(
i
π
τ
)
]
=
∑
n
=
−
∞
∞
exp
[
i
π
τ
(
n
+
1
/
2
)
2
]
{\displaystyle \vartheta _{10}[\exp(i\pi \tau )]=\sum _{n=-\infty }^{\infty }\exp[i\pi \tau (n+1/2)^{2}]}
ϑ
00
[
exp
(
i
π
τ
)
]
=
∑
n
=
−
∞
∞
exp
(
i
π
τ
n
2
)
{\displaystyle \vartheta _{00}[\exp(i\pi \tau )]=\sum _{n=-\infty }^{\infty }\exp(i\pi \tau n^{2})}
Die Kongruenzuntergruppe Γ(2) ist hierbei folgendermaßen beschaffen:
Γ
(
2
)
:=
{
(
a
b
c
d
)
∈
SL
2
(
Z
)
|
a
≡
d
≡
1
(
mod
2
)
,
b
≡
c
≡
0
(
mod
2
)
}
=
⟨
(
1
2
0
1
)
,
(
1
0
2
1
)
,
(
−
1
0
0
−
1
)
⟩
{\displaystyle \Gamma (2):=\left\{{\begin{pmatrix}a&b\\c&d\end{pmatrix}}\in \operatorname {SL} _{2}(\mathbb {Z} )|a\equiv d\equiv 1{\pmod {2}},b\equiv c\equiv 0{\pmod {2}}\right\}=\left\langle {\begin{pmatrix}1&2\\0&1\end{pmatrix}},{\begin{pmatrix}1&0\\2&1\end{pmatrix}},{\begin{pmatrix}-1&0\\0&-1\end{pmatrix}}\right\rangle }
Ausdruck über die Dedekindsche Etafunktion :
λ
(
τ
)
=
16
η
8
(
τ
/
2
)
η
16
(
2
τ
)
η
24
(
τ
)
{\displaystyle \lambda (\tau )={\frac {16\eta ^{8}(\tau /2)\eta ^{16}(2\tau )}{\eta ^{24}(\tau )}}}
Ausdruck über die Weierstraß-Funktion :[ 1]
λ
(
τ
)
=
℘
(
τ
/
2
+
1
/
2
,
τ
)
−
℘
(
τ
/
2
,
τ
)
℘
(
1
/
2
,
τ
)
−
℘
(
τ
/
2
,
τ
)
{\displaystyle \lambda (\tau )={\frac {\wp (\tau /2+1/2,\tau )-\wp (\tau /2,\tau )}{\wp (1/2,\tau )-\wp (\tau /2,\tau )}}}
Funktionsgraph für λ(ix) = λ*(x²)², das Quadrat der Lambda-Stern-Funktion Die Elliptische Lambda-Funktion ausgedrückt mit einem Stern oben rechts über dem Lambda liefert den elliptischen Modul beziehungsweise die Exzentrizität auf folgende Weise:
K
[
1
−
λ
∗
(
x
)
2
]
/
K
[
λ
∗
(
x
)
]
=
x
{\displaystyle K\left[{\sqrt {1-\lambda ^{*}(x)^{2}}}\right]/K[\lambda ^{*}(x)]={\sqrt {x}}}
Dabei bezeichnet K das vollständige elliptische Integral erster Art.
Die Funktionen Lambda und Lambda-Stern stehen in folgender Beziehung zueinander:
λ
∗
(
x
)
=
λ
(
i
x
)
{\displaystyle \lambda ^{*}(x)={\sqrt {\lambda (i{\sqrt {x}})}}}
Primär ist die Funktion λ*(x) so über die Theta-Nullwertfunktionen definiert:
λ
∗
(
x
)
=
ϑ
10
2
[
exp
(
−
π
x
)
]
ϑ
00
2
[
exp
(
−
π
x
)
]
{\displaystyle \lambda ^{*}(x)={\frac {\vartheta _{10}^{2}[\exp(-\pi {\sqrt {x}})]}{\vartheta _{00}^{2}[\exp(-\pi {\sqrt {x}})]}}}
Ebenso kann Lambda-Stern-Funktion über den pythagoräisch komplementären Modul dargestellt werden:
λ
∗
(
x
)
=
ϑ
01
2
[
exp
(
−
π
/
x
)
]
ϑ
00
2
[
exp
(
−
π
/
x
)
]
{\displaystyle \lambda ^{*}(x)={\frac {\vartheta _{01}^{2}[\exp(-\pi /{\sqrt {x}})]}{\vartheta _{00}^{2}[\exp(-\pi /{\sqrt {x}})]}}}
Auch über die Theta-Nicht-Nullwertfunktionen ist die Definition möglich:
λ
∗
(
x
)
=
ϑ
10
[
1
4
π
;
exp
(
−
1
2
π
x
)
]
4
ϑ
00
[
1
4
π
;
exp
(
−
1
2
π
x
)
]
4
{\displaystyle \lambda ^{*}(x)={\frac {\vartheta _{10}\left[{\tfrac {1}{4}}\pi ;\exp \left(-{\tfrac {1}{2}}\pi {\sqrt {x}}\right)\right]^{4}}{\vartheta _{00}\left[{\tfrac {1}{4}}\pi ;\exp \left(-{\tfrac {1}{2}}\pi {\sqrt {x}}\right)\right]^{4}}}}
Die Thetafunktionen selbst sind nach Whittaker und Watson so definiert:
ϑ
00
(
v
;
w
)
=
∏
n
=
1
∞
(
1
−
w
2
n
)
[
1
+
2
cos
(
2
v
)
w
2
n
−
1
+
w
4
n
−
2
]
{\displaystyle \vartheta _{00}(v;w)=\prod _{n=1}^{\infty }\left(1-w^{2n}\right)\left[1+2\cos(2v)w^{2n-1}+w^{4n-2}\right]}
ϑ
01
(
v
;
w
)
=
∏
n
=
1
∞
(
1
−
w
2
n
)
[
1
−
2
cos
(
2
v
)
w
2
n
−
1
+
w
4
n
−
2
]
{\displaystyle \vartheta _{01}(v;w)=\prod _{n=1}^{\infty }\left(1-w^{2n}\right)\left[1-2\cos(2v)w^{2n-1}+w^{4n-2}\right]}
ϑ
10
(
v
;
w
)
=
2
w
1
/
4
cos
(
v
)
∏
n
=
1
∞
(
1
−
w
2
n
)
[
1
+
2
cos
(
2
v
)
w
2
n
+
w
4
n
]
{\displaystyle \vartheta _{10}(v;w)=2w^{1/4}\cos(v)\prod _{n=1}^{\infty }\left(1-w^{2n}\right)\left[1+2\cos(2v)w^{2n}+w^{4n}\right]}
Außerdem gelten folgende Ausdrucksweisen:
ϑ
00
(
w
)
=
ϑ
00
(
0
;
w
)
{\displaystyle \vartheta _{00}(w)=\vartheta _{00}(0;w)}
ϑ
01
(
w
)
=
ϑ
01
(
0
;
w
)
{\displaystyle \vartheta _{01}(w)=\vartheta _{01}(0;w)}
ϑ
10
(
w
)
=
ϑ
10
(
0
;
w
)
{\displaystyle \vartheta _{10}(w)=\vartheta _{10}(0;w)}
Die Lambda-Stern-Werte können mit diesen sehr schnell konvergierenden Definitionsformeln[ 2]
λ
∗
(
x
)
=
{
∑
a
=
−
∞
∞
exp
[
−
(
a
+
1
/
2
)
2
π
x
]
}
2
{
∑
a
=
−
∞
∞
exp
(
−
a
2
π
x
)
}
−
2
{\displaystyle \lambda ^{*}(x)={\biggl \{}\sum _{a=-\infty }^{\infty }\exp {\bigl [}-(a+1/2)^{2}\pi {\sqrt {x}}{\bigr ]}{\biggr \}}^{2}{\biggl \{}\sum _{a=-\infty }^{\infty }\exp(-a^{2}\pi {\sqrt {x}}){\biggr \}}^{-2}}
λ
∗
(
x
)
=
{
∑
a
=
−
∞
∞
sech
[
(
a
+
1
/
2
)
π
x
]
}
{
∑
a
=
−
∞
∞
sech
(
a
π
x
)
}
−
1
{\displaystyle \lambda ^{*}(x)={\biggl \{}\sum _{a=-\infty }^{\infty }\operatorname {sech} {\bigl [}(a+1/2)\pi {\sqrt {x}}{\bigr ]}{\biggr \}}{\biggl \{}\sum _{a=-\infty }^{\infty }\operatorname {sech} (a\pi {\sqrt {x}}){\biggr \}}^{-1}}
λ
∗
(
x
)
=
[
∑
a
=
−
∞
∞
(
−
1
)
a
exp
(
−
a
2
π
x
)
]
2
[
∑
a
=
−
∞
∞
exp
(
−
a
2
π
x
)
]
−
2
{\displaystyle \lambda ^{*}(x)={\biggl [}\sum _{a=-\infty }^{\infty }(-1)^{a}\exp {\biggl (}-{\frac {a^{2}\pi }{\sqrt {x}}}{\biggr )}{\biggr ]}^{2}{\biggl [}\sum _{a=-\infty }^{\infty }\exp \left(-{\frac {a^{2}\pi }{\sqrt {x}}}\right){\biggr ]}^{-2}}
λ
∗
(
x
)
=
∏
a
=
0
∞
tanh
[
(
a
+
1
/
2
)
π
x
]
4
{\displaystyle \lambda ^{*}(x)=\prod _{a=0}^{\infty }\operatorname {tanh} {\biggl [}{\frac {(a+1/2)\pi }{\sqrt {x}}}{\biggr ]}^{4}}
Die Jacobische Theta-Nullwertfunktion ϑ₀₀ hat diese Integralidentität:
ϑ
00
[
exp
(
−
π
x
)
]
=
1
+
2
exp
(
−
π
x
)
∫
0
∞
exp
(
−
π
y
2
)
exp
(
2
π
x
)
−
cos
(
2
π
x
4
y
)
cosh
(
2
π
x
)
−
cos
(
2
π
x
4
y
)
d
y
{\displaystyle \vartheta _{00}[\exp(-\pi {\sqrt {x}})]=1+2\exp(-\pi {\sqrt {x}})\int _{0}^{\infty }\exp \left(-\pi y^{2}\right){\frac {\exp(2\pi {\sqrt {x}})-\cos(2\pi {\sqrt[{4}]{x}}\,y)}{\cosh(2\pi {\sqrt {x}})-\cos(2\pi {\sqrt[{4}]{x}}\,y)}}\mathrm {d} y}
ϑ
00
[
exp
(
−
2
π
x
)
]
=
1
+
2
exp
(
−
2
π
x
)
∫
0
∞
exp
(
−
π
y
2
)
exp
(
4
π
x
)
−
cos
(
2
π
4
x
4
y
)
cosh
(
4
π
x
)
−
cos
(
2
π
4
x
4
y
)
d
y
{\displaystyle \vartheta _{00}[\exp(-2\pi {\sqrt {x}})]=1+2\exp(-2\pi {\sqrt {x}})\int _{0}^{\infty }\exp \left(-\pi y^{2}\right){\frac {\exp(4\pi {\sqrt {x}})-\cos(2\pi {\sqrt[{4}]{4x}}\,y)}{\cosh(4\pi {\sqrt {x}})-\cos(2\pi {\sqrt[{4}]{4x}}\,y)}}\mathrm {d} y}
Die Lambda-Stern-Funktion kann dann auf jenem Definitionsweg dargestellt werden:
λ
∗
(
x
)
=
sin
⟨
2
arccos
{
ϑ
00
[
exp
(
−
2
π
x
)
]
ϑ
00
[
exp
(
−
π
x
)
]
}
⟩
{\displaystyle \lambda ^{*}(x)=\sin {\biggl \langle }2\arccos {\biggl \{}{\frac {\vartheta _{00}[\exp(-2\pi {\sqrt {x}})]}{\vartheta _{00}[\exp(-\pi {\sqrt {x}})]}}{\biggr \}}{\biggr \rangle }}
Für die Thetafunktionen ϑ₁₀ und ϑ₀₀ in reeller Form gelten folgende Formeln:
ϑ
10
[
exp
(
−
π
x
)
]
=
∑
a
=
−
∞
∞
exp
[
−
(
a
+
1
2
)
2
π
x
]
=
{
∑
a
=
−
∞
∞
sech
[
(
a
+
1
2
)
π
x
]
}
1
/
2
=
{\displaystyle \vartheta _{10}[\exp(-\pi {\sqrt {x}})]=\sum _{a=-\infty }^{\infty }\exp \left[-\left(a+{\tfrac {1}{2}}\right)^{2}\pi {\sqrt {x}}\right]={\biggl \{}\sum _{a=-\infty }^{\infty }\operatorname {sech} \left[\left(a+{\tfrac {1}{2}}\right)\pi {\sqrt {x}}\right]{\biggr \}}^{1/2}=}
=
2
π
−
1
λ
∗
(
x
)
K
[
λ
∗
(
x
)
]
=
λ
∗
(
4
x
)
4
4
π
−
1
K
[
λ
∗
(
4
x
)
]
{\displaystyle ={\sqrt {2\pi ^{-1}\lambda ^{*}(x)K[\lambda ^{*}(x)]}}={\sqrt[{4}]{\lambda ^{*}(4x)}}{\sqrt {4\pi ^{-1}K[\lambda ^{*}(4x)]}}}
ϑ
00
[
exp
(
−
π
x
)
]
=
∑
a
=
−
∞
∞
exp
(
−
a
2
π
x
)
=
[
∑
a
=
−
∞
∞
sech
(
a
π
x
)
]
1
/
2
=
2
π
−
1
K
[
λ
∗
(
x
)
]
=
agm
[
1
;
λ
∗
(
1
/
x
)
]
−
1
/
2
{\displaystyle \vartheta _{00}[\exp(-\pi {\sqrt {x}})]=\sum _{a=-\infty }^{\infty }\exp(-a^{2}\pi {\sqrt {x}})={\biggl [}\sum _{a=-\infty }^{\infty }\operatorname {sech} (a\pi {\sqrt {x}}){\biggr ]}^{1/2}={\sqrt {2\pi ^{-1}K[\lambda ^{*}(x)]}}=\operatorname {agm} [1;\lambda ^{*}(1/x)]^{-1/2}}
ϑ
01
[
exp
(
−
π
x
)
]
=
∑
a
=
−
∞
∞
(
−
1
)
a
exp
(
−
a
2
π
x
)
=
2
π
−
1
λ
∗
(
1
/
x
)
K
[
λ
∗
(
x
)
]
{\displaystyle \vartheta _{01}[\exp(-\pi {\sqrt {x}})]=\sum _{a=-\infty }^{\infty }(-1)^{a}\exp(-a^{2}\pi {\sqrt {x}})={\sqrt {2\pi ^{-1}\lambda ^{*}(1/x)K[\lambda ^{*}(x)]}}}
Mit der Abkürzung agm wird das arithmetisch geometrische Mittel zum Ausdrück gebracht.
Von diesen beiden Thetafunktionen werden im Folgenden einige Theta-Nullwerte aufgelistet.
Unendliche Summe der Kehrwerte ungeradstelliger Fibonacci-Zahlen :
∑
n
=
1
∞
1
F
2
n
−
1
=
∑
n
=
1
∞
5
2
sech
[
(
n
−
1
2
)
arcosh
(
3
2
)
]
=
5
4
∑
a
=
−
∞
∞
sech
[
(
a
+
1
2
)
arcosh
(
3
2
)
]
=
{\displaystyle \sum _{n=1}^{\infty }{\frac {1}{F_{2n-1}}}=\sum _{n=1}^{\infty }{\frac {\sqrt {5}}{2}}\operatorname {sech} \left[\left(n-{\tfrac {1}{2}}\right)\operatorname {arcosh} \left({\tfrac {3}{2}}\right)\right]={\frac {\sqrt {5}}{4}}\sum _{a=-\infty }^{\infty }\operatorname {sech} \left[\left(a+{\tfrac {1}{2}}\right)\operatorname {arcosh} \left({\tfrac {3}{2}}\right)\right]=}
=
5
4
ϑ
10
(
Φ
−
2
)
2
=
5
8
[
ϑ
00
(
Φ
−
1
)
2
−
ϑ
01
(
Φ
−
1
)
2
]
=
5
π
λ
∗
[
16
π
−
2
ln
(
Φ
)
2
]
K
{
λ
∗
[
16
π
−
2
ln
(
Φ
)
2
]
}
{\displaystyle ={\frac {\sqrt {5}}{4}}\vartheta _{10}(\Phi ^{-2})^{2}={\frac {\sqrt {5}}{8}}{\bigl [}\vartheta _{00}(\Phi ^{-1})^{2}-\vartheta _{01}(\Phi ^{-1})^{2}{\bigr ]}={\frac {\sqrt {5}}{\pi }}{\sqrt {\lambda ^{*}[16\,\pi ^{-2}\ln(\Phi )^{2}]}}K\{\lambda ^{*}[16\,\pi ^{-2}\ln(\Phi )^{2}]\}}
Dabei ist
Φ
=
5
+
1
2
{\displaystyle \Phi ={\tfrac {{\sqrt {5}}+1}{2}}}
goldene Zahl .
Unendliche Summe der Kehrwerte ungeradstelliger Pell-Zahlen :
∑
n
=
1
∞
1
P
(
2
n
−
1
)
=
2
2
π
λ
∗
[
16
π
−
2
arsinh
(
1
)
2
]
K
{
λ
∗
[
16
π
−
2
arsinh
(
1
)
2
]
}
{\displaystyle \sum _{n=1}^{\infty }{\frac {1}{P(2n-1)}}={\frac {2{\sqrt {2}}}{\pi }}{\sqrt {\lambda ^{*}[16\,\pi ^{-2}\operatorname {arsinh} (1)^{2}]}}K\{\lambda ^{*}[16\,\pi ^{-2}\operatorname {arsinh} (1)^{2}]\}}
Die Funktion
λ
(
τ
)
{\displaystyle \lambda (\tau )}
τ
⟼
τ
+
2
;
τ
⟼
τ
1
−
2
τ
{\displaystyle \tau \longmapsto \tau +2;\tau \longmapsto {\frac {\tau }{1-2\tau }}}
Die Erzeuger der modularen Gruppen sind wie folgt beschaffen:
τ
⟼
τ
+
1
;
λ
⟼
λ
λ
−
1
{\displaystyle \tau \longmapsto \tau +1;\lambda \longmapsto {\frac {\lambda }{\lambda -1}}}
τ
⟼
−
1
τ
;
λ
⟼
1
−
λ
{\displaystyle \tau \longmapsto -{\frac {1}{\tau }};\lambda \longmapsto 1-\lambda }
Folglich verhält sich die Gruppe in Bezug auf
λ
(
τ
)
{\displaystyle \lambda (\tau )}
Das Doppelverhältnis weist folgende sechs Werte auf:
{
λ
,
1
1
−
λ
,
λ
−
1
λ
,
1
λ
,
λ
λ
−
1
,
1
−
λ
}
{\displaystyle \left\{\lambda ,{\frac {1}{1-\lambda }},{\frac {\lambda -1}{\lambda }},{\frac {1}{\lambda }},{\frac {\lambda }{\lambda -1}},1-\lambda \right\}}
Generell ist jeder Lambda-Stern-Wert einer positiven rationalen Zahl eine positive algebraische Zahl:
λ
∗
(
x
∈
Q
+
)
∈
A
+
{\displaystyle \lambda ^{*}(x\in \mathbb {Q} ^{+})\in \mathbb {A} ^{+}}
Folgende Beziehung gilt für alle n ∈ ℕ:
n
=
∑
a
=
1
n
dn
{
2
a
n
K
[
λ
∗
(
1
n
)
]
;
λ
∗
(
1
n
)
}
{\displaystyle {\sqrt {n}}=\sum _{a=1}^{n}\operatorname {dn} {\biggl \{}{\frac {2a}{n}}K{\biggl [}\lambda ^{*}{\biggl (}{\frac {1}{n}}{\biggr )}{\biggr ]};\lambda ^{*}{\biggl (}{\frac {1}{n}}{\biggr )}{\biggr \}}}
Hierbei ist dn die Jacobische elliptische Funktion Delta Amplitudinis.
Weiterhin gilt für alle Zahlen n ∈ ℕ:
λ
∗
(
n
2
x
)
=
λ
∗
(
x
)
n
∏
a
=
1
n
sn
{
2
a
−
1
n
K
[
λ
∗
(
x
)
]
;
λ
∗
(
x
)
}
2
{\displaystyle \lambda ^{*}(n^{2}x)=\lambda ^{*}(x)^{n}\prod _{a=1}^{n}\operatorname {sn} {\biggl \{}{\frac {2a-1}{n}}K{\bigl [}\lambda ^{*}(x){\bigr ]};\lambda ^{*}(x){\biggr \}}^{2}}
λ
∗
(
4
n
2
)
=
∏
a
=
1
n
sl
(
2
a
−
1
4
n
ϖ
)
4
{\displaystyle \lambda ^{*}(4n^{2})=\prod _{a=1}^{n}\operatorname {sl} \left({\frac {2a-1}{4n}}\varpi \right)^{4}}
Hierbei ist sn die Jacobische elliptische Funktion Sinus Amplitudinis, während sl der lemniskatische Sinus ist.
Folgende weitere Beziehungen[ 3]
λ
∗
(
x
)
2
+
λ
∗
(
1
/
x
)
2
=
1
{\displaystyle \lambda ^{*}(x)^{2}+\lambda ^{*}(1/x)^{2}=1}
[
λ
∗
(
x
)
+
1
]
[
λ
∗
(
4
/
x
)
+
1
]
=
2
{\displaystyle [\lambda ^{*}(x)+1][\lambda ^{*}(4/x)+1]=2}
λ
∗
(
4
x
)
=
1
−
1
−
λ
∗
(
x
)
2
1
+
1
−
λ
∗
(
x
)
2
=
λ
∗
(
x
)
2
[
1
+
1
−
λ
∗
(
x
)
2
]
2
{\displaystyle \lambda ^{*}(4x)={\frac {1-{\sqrt {1-\lambda ^{*}(x)^{2}}}}{1+{\sqrt {1-\lambda ^{*}(x)^{2}}}}}={\frac {\lambda ^{*}(x)^{2}}{{\bigl [}\,1+{\sqrt {1-\lambda ^{*}(x)^{2}}}\,{\bigr ]}^{2}}}}
λ
∗
(
4
x
)
=
tan
{
1
2
arcsin
[
λ
∗
(
x
)
]
}
2
=
tanh
{
1
2
artanh
[
λ
∗
(
x
)
]
}
2
{\displaystyle \lambda ^{*}(4x)=\tan \left\{{\tfrac {1}{2}}\arcsin[\lambda ^{*}(x)]\right\}^{2}=\tanh \left\{{\tfrac {1}{2}}\operatorname {artanh} [\lambda ^{*}(x)]\right\}^{2}}
[
λ
∗
(
x
)
−
λ
∗
(
9
x
)
]
4
=
16
λ
∗
(
x
)
λ
∗
(
9
x
)
[
1
−
λ
∗
(
x
)
2
]
[
1
−
λ
∗
(
9
x
)
2
]
{\displaystyle [\lambda ^{*}(x)-\lambda ^{*}(9x)]^{4}=16\lambda ^{*}(x)\lambda ^{*}(9x)\left[1-\lambda ^{*}(x)^{2}\right]\left[1-\lambda ^{*}(9x)^{2}\right]}
λ
∗
(
x
)
−
λ
∗
(
9
x
)
=
2
[
λ
∗
(
x
)
λ
∗
(
9
x
)
]
1
/
4
−
2
[
λ
∗
(
x
)
λ
∗
(
9
x
)
]
3
/
4
{\displaystyle \lambda ^{*}(x)-\lambda ^{*}(9x)=2[\lambda ^{*}(x)\lambda ^{*}(9x)]^{1/4}-2[\lambda ^{*}(x)\lambda ^{*}(9x)]^{3/4}}
λ
∗
(
x
)
λ
∗
(
9
x
)
+
[
1
−
λ
∗
(
x
)
2
]
[
1
−
λ
∗
(
9
x
)
2
]
4
=
1
{\displaystyle {\sqrt {\lambda ^{*}(x)\lambda ^{*}(9x)}}+{\sqrt[{4}]{\left[1-\lambda ^{*}(x)^{2}\right]\left[1-\lambda ^{*}(9x)^{2}\right]}}=1}
[
2
λ
∗
(
x
)
1
−
λ
∗
(
x
)
2
]
1
/
2
−
[
2
λ
∗
(
25
x
)
1
−
λ
∗
(
25
x
)
2
]
1
/
2
=
2
[
2
λ
∗
(
x
)
1
−
λ
∗
(
x
)
2
]
1
/
12
[
2
λ
∗
(
25
x
)
1
−
λ
∗
(
25
x
)
2
]
1
/
12
+
2
[
2
λ
∗
(
x
)
1
−
λ
∗
(
x
)
2
]
5
/
12
[
2
λ
∗
(
25
x
)
1
−
λ
∗
(
25
x
)
2
]
5
/
12
{\displaystyle {\biggl [}{\frac {2\lambda ^{*}(x)}{1-\lambda ^{*}(x)^{2}}}{\biggr ]}^{1/2}-{\biggl [}{\frac {2\lambda ^{*}(25x)}{1-\lambda ^{*}(25x)^{2}}}{\biggr ]}^{1/2}=2{\biggl [}{\frac {2\lambda ^{*}(x)}{1-\lambda ^{*}(x)^{2}}}{\biggr ]}^{1/12}{\biggl [}{\frac {2\lambda ^{*}(25x)}{1-\lambda ^{*}(25x)^{2}}}{\biggr ]}^{1/12}+2{\biggl [}{\frac {2\lambda ^{*}(x)}{1-\lambda ^{*}(x)^{2}}}{\biggr ]}^{5/12}{\biggl [}{\frac {2\lambda ^{*}(25x)}{1-\lambda ^{*}(25x)^{2}}}{\biggr ]}^{5/12}}
[
λ
∗
(
x
)
1
/
2
−
λ
∗
(
25
x
)
1
/
2
]
[
λ
∗
(
x
)
+
λ
∗
(
25
x
)
+
6
λ
∗
(
x
)
1
/
2
λ
∗
(
25
x
)
1
/
2
]
=
4
λ
∗
(
x
)
1
/
4
λ
∗
(
25
x
)
1
/
4
[
1
−
λ
∗
(
x
)
λ
∗
(
25
x
)
]
{\displaystyle \left[\lambda ^{*}(x)^{1/2}-\lambda ^{*}(25x)^{1/2}\right]\left[\lambda ^{*}(x)+\lambda ^{*}(25x)+6\lambda ^{*}(x)^{1/2}\lambda ^{*}(25x)^{1/2}\right]=4\lambda ^{*}(x)^{1/4}\lambda ^{*}(25x)^{1/4}[1-\lambda ^{*}(x)\lambda ^{*}(25x)]}
λ
∗
(
x
)
λ
∗
(
25
x
)
+
[
1
−
λ
∗
(
x
)
]
2
[
1
−
λ
∗
(
25
x
)
]
2
+
2
16
λ
∗
(
x
)
2
λ
∗
(
25
x
)
2
[
1
−
λ
∗
(
x
)
2
]
[
1
−
λ
∗
(
25
x
)
2
]
6
=
1
{\displaystyle \lambda ^{*}(x)\lambda ^{*}(25x)+{\sqrt {[1-\lambda ^{*}(x)]^{2}[1-\lambda ^{*}(25x)]^{2}}}+2{\sqrt[{6}]{16\lambda ^{*}(x)^{2}\lambda ^{*}(25x)^{2}\left[1-\lambda ^{*}(x)^{2}\right]\left[1-\lambda ^{*}(25x)^{2}\right]}}=1}
λ
∗
(
x
)
λ
∗
(
49
x
)
4
+
[
1
−
λ
∗
(
x
)
2
]
[
1
−
λ
∗
(
49
x
)
2
]
8
=
1
{\displaystyle {\sqrt[{4}]{\lambda ^{*}(x)\lambda ^{*}(49x)}}+{\sqrt[{8}]{\left[1-\lambda ^{*}(x)^{2}\right]\left[1-\lambda ^{*}(49x)^{2}\right]}}=1}
λ
∗
(
x
)
λ
∗
(
121
x
)
+
[
1
−
λ
∗
(
x
)
2
]
[
1
−
λ
∗
(
121
x
)
2
]
4
+
2
16
λ
∗
(
x
)
2
λ
∗
(
121
x
)
2
[
1
−
λ
∗
(
x
)
2
]
[
1
−
λ
∗
(
121
x
)
2
]
12
=
1
{\displaystyle {\sqrt {\lambda ^{*}(x)\lambda ^{*}(121x)}}+{\sqrt[{4}]{\left[1-\lambda ^{*}(x)^{2}\right]\left[1-\lambda ^{*}(121x)^{2}\right]}}+2{\sqrt[{12}]{16\lambda ^{*}(x)^{2}\lambda ^{*}(121x)^{2}\left[1-\lambda ^{*}(x)^{2}\right]\left[1-\lambda ^{*}(121x)^{2}\right]}}=1}
λ
∗
(
x
)
λ
∗
(
529
x
)
4
+
[
1
−
λ
∗
(
x
)
2
]
[
1
−
λ
∗
(
529
x
)
2
]
8
+
2
16
λ
∗
(
x
)
2
λ
∗
(
529
x
)
2
[
1
−
λ
∗
(
x
)
2
]
[
1
−
λ
∗
(
529
x
)
2
]
24
=
1
{\displaystyle {\sqrt[{4}]{\lambda ^{*}(x)\lambda ^{*}(529x)}}+{\sqrt[{8}]{\left[1-\lambda ^{*}(x)^{2}\right]\left[1-\lambda ^{*}(529x)^{2}\right]}}+{\sqrt {2}}{\sqrt[{24}]{16\lambda ^{*}(x)^{2}\lambda ^{*}(529x)^{2}\left[1-\lambda ^{*}(x)^{2}\right]\left[1-\lambda ^{*}(529x)^{2}\right]}}=1}
Folgende Beziehungen gelten zu den Ramanujanschen Funktionen g und G:
G
(
x
)
=
sin
{
2
arcsin
[
λ
∗
(
x
)
]
}
−
1
/
12
=
1
/
[
2
λ
∗
(
x
)
12
1
−
λ
∗
(
x
)
2
24
]
{\displaystyle G(x)=\sin\{2\arcsin[\lambda ^{*}(x)]\}^{-1/12}=1\left/\left[{\sqrt[{12}]{2\lambda ^{*}(x)}}{\sqrt[{24}]{1-\lambda ^{*}(x)^{2}}}\right]\right.}
g
(
x
)
=
tan
{
2
arctan
[
λ
∗
(
x
)
]
}
−
1
/
12
=
[
1
−
λ
∗
(
x
)
2
]
/
[
2
λ
∗
(
x
)
]
12
{\displaystyle g(x)=\tan\{2\arctan[\lambda ^{*}(x)]\}^{-1/12}={\sqrt[{12}]{[1-\lambda ^{*}(x)^{2}]/[2\lambda ^{*}(x)]}}}
λ
∗
(
x
)
=
tan
{
1
2
arctan
[
g
(
x
)
−
12
]
}
=
g
(
x
)
24
+
1
−
g
(
x
)
12
{\displaystyle \lambda ^{*}(x)=\tan \left\{{\tfrac {1}{2}}\arctan \left[g(x)^{-12}\right]\right\}={\sqrt {g(x)^{24}+1}}-g(x)^{12}}
In dieser Liste werden die Lambda-Stern-Werte[ 4]
λ
∗
(
1
)
=
1
2
2
{\displaystyle \lambda ^{*}(1)={\frac {1}{2}}{\sqrt {2}}}
λ
∗
(
2
)
=
2
−
1
{\displaystyle \lambda ^{*}(2)={\sqrt {2}}-1}
λ
∗
(
3
)
=
1
4
(
6
−
2
)
{\displaystyle \lambda ^{*}(3)={\frac {1}{4}}\left({\sqrt {6}}-{\sqrt {2}}\right)}
λ
∗
(
4
)
=
(
2
−
1
)
2
{\displaystyle \lambda ^{*}(4)=\left({\sqrt {2}}-1\right)^{2}}
λ
∗
(
5
)
=
1
2
(
5
−
1
−
3
−
5
)
{\displaystyle \lambda ^{*}(5)={\frac {1}{2}}\left({\sqrt {{\sqrt {5}}-1}}-{\sqrt {3-{\sqrt {5}}}}\right)}
λ
∗
(
6
)
=
(
2
−
3
)
(
3
−
2
)
{\displaystyle \lambda ^{*}(6)=\left(2-{\sqrt {3}}\right)\left({\sqrt {3}}-{\sqrt {2}}\right)}
λ
∗
(
7
)
=
1
8
(
3
2
−
14
)
{\displaystyle \lambda ^{*}(7)={\frac {1}{8}}\left(3{\sqrt {2}}-{\sqrt {14}}\right)}
λ
∗
(
8
)
=
(
2
+
1
−
2
2
+
2
)
2
{\displaystyle \lambda ^{*}(8)=\left({\sqrt {2}}+1-{\sqrt {2{\sqrt {2}}+2}}\right)^{2}}
λ
∗
(
9
)
=
1
2
(
3
−
1
)
(
2
−
3
4
)
{\displaystyle \lambda ^{*}(9)={\frac {1}{2}}\left({\sqrt {3}}-1\right)\left({\sqrt {2}}-{\sqrt[{4}]{3}}\right)}
λ
∗
(
10
)
=
(
10
−
3
)
(
2
−
1
)
2
{\displaystyle \lambda ^{*}(10)=\left({\sqrt {10}}-3\right)\left({\sqrt {2}}-1\right)^{2}}
λ
∗
(
11
)
=
1
16
(
22
+
3
2
)
(
1
3
6
3
+
2
11
3
−
1
3
6
3
−
2
11
3
+
1
3
11
−
1
)
4
{\displaystyle \lambda ^{*}(11)={\frac {1}{16}}\left({\sqrt {22}}+3{\sqrt {2}}\right)\left({\frac {1}{3}}{\sqrt[{3}]{6{\sqrt {3}}+2{\sqrt {11}}}}-{\frac {1}{3}}{\sqrt[{3}]{6{\sqrt {3}}-2{\sqrt {11}}}}+{\frac {1}{3}}{\sqrt {11}}-1\right)^{4}}
λ
∗
(
12
)
=
(
3
−
2
)
2
(
2
−
1
)
2
{\displaystyle \lambda ^{*}(12)=\left({\sqrt {3}}-{\sqrt {2}}\right)^{2}\left({\sqrt {2}}-1\right)^{2}}
λ
∗
(
13
)
=
1
2
(
5
13
−
17
−
19
−
5
13
)
{\displaystyle \lambda ^{*}(13)={\frac {1}{2}}\left({\sqrt {5{\sqrt {13}}-17}}-{\sqrt {19-5{\sqrt {13}}}}\right)}
λ
∗
(
14
)
=
(
2
2
+
2
−
8
2
+
11
)
(
8
2
+
11
−
8
2
+
10
)
{\displaystyle \lambda ^{*}(14)=\left(2{\sqrt {2}}+2-{\sqrt {8{\sqrt {2}}+11}}\right)\left({\sqrt {8{\sqrt {2}}+11}}-{\sqrt {8{\sqrt {2}}+10}}\right)}
λ
∗
(
15
)
=
1
16
(
10
−
6
)
(
3
−
5
)
(
2
−
3
)
{\displaystyle \lambda ^{*}(15)={\frac {1}{16}}\left({\sqrt {10}}-{\sqrt {6}}\right)\left(3-{\sqrt {5}}\right)\left(2-{\sqrt {3}}\right)}
λ
∗
(
16
)
=
(
2
+
1
)
2
(
2
4
−
1
)
4
{\displaystyle \lambda ^{*}(16)=\left({\sqrt {2}}+1\right)^{2}\left({\sqrt[{4}]{2}}-1\right)^{4}}
λ
∗
(
17
)
=
1
4
(
7
+
17
−
17
+
3
)
(
5
+
17
−
10
17
+
38
4
)
{\displaystyle \lambda ^{*}(17)={\frac {1}{4}}\left({\sqrt {7+{\sqrt {17}}}}-{\sqrt {{\sqrt {17}}+3}}\right)\left({\sqrt {5+{\sqrt {17}}}}-{\sqrt[{4}]{10{\sqrt {17}}+38}}\right)}
λ
∗
(
18
)
=
(
2
−
3
)
2
(
2
−
1
)
3
{\displaystyle \lambda ^{*}(18)=\left(2-{\sqrt {3}}\right)^{2}\left({\sqrt {2}}-1\right)^{3}}
λ
∗
(
19
)
=
1
16
(
3
38
+
13
2
)
[
1
6
(
19
−
2
+
3
)
3
3
−
19
3
−
1
6
(
19
−
2
−
3
)
3
3
+
19
3
−
1
3
(
5
−
19
)
]
4
{\displaystyle \lambda ^{*}(19)={\frac {1}{16}}\left(3{\sqrt {38}}+13{\sqrt {2}}\right)\left[{\frac {1}{6}}\left({\sqrt {19}}-2+{\sqrt {3}}\right){\sqrt[{3}]{3{\sqrt {3}}-{\sqrt {19}}}}-{\frac {1}{6}}\left({\sqrt {19}}-2-{\sqrt {3}}\right){\sqrt[{3}]{3{\sqrt {3}}+{\sqrt {19}}}}-{\frac {1}{3}}\left(5-{\sqrt {19}}\right)\right]^{4}}
λ
∗
(
20
)
=
(
10
−
3
)
(
5
+
2
)
(
2
−
1
)
(
5
−
1
−
1
)
2
{\displaystyle \lambda ^{*}(20)=\left({\sqrt {10}}-3\right)\left({\sqrt {5}}+2\right)\left({\sqrt {2}}-1\right)\left({\sqrt {{\sqrt {5}}-1}}-1\right)^{2}}
λ
∗
(
21
)
=
1
4
[
(
7
−
3
)
(
3
+
1
)
7
−
2
−
2
14
+
5
2
−
42
+
2
6
]
{\displaystyle \lambda ^{*}(21)={\frac {1}{4}}\left[\left({\sqrt {7}}-{\sqrt {3}}\right)\left({\sqrt {3}}+1\right){\sqrt {{\sqrt {7}}-2}}-2{\sqrt {14}}+5{\sqrt {2}}-{\sqrt {42}}+2{\sqrt {6}}\right]}
λ
∗
(
22
)
=
(
10
−
3
11
)
(
3
11
−
7
2
)
{\displaystyle \lambda ^{*}(22)=\left(10-3{\sqrt {11}}\right)\left(3{\sqrt {11}}-7{\sqrt {2}}\right)}
λ
∗
(
23
)
=
1
32
(
5
2
+
46
)
[
2
3
+
1
6
(
3
+
1
)
100
−
12
69
3
−
1
6
(
3
−
1
)
100
+
12
69
3
]
4
{\displaystyle \lambda ^{*}(23)={\frac {1}{32}}\left(5{\sqrt {2}}+{\sqrt {46}}\right)\left[{\frac {2}{3}}+{\frac {1}{6}}\left({\sqrt {3}}+1\right){\sqrt[{3}]{100-12{\sqrt {69}}}}-{\frac {1}{6}}\left({\sqrt {3}}-1\right){\sqrt[{3}]{100+12{\sqrt {69}}}}\right]^{4}}
λ
∗
(
24
)
=
(
2
+
3
)
2
[
3
+
2
−
(
3
−
1
)
(
2
+
1
)
3
+
2
]
2
{\displaystyle \lambda ^{*}(24)=\left(2+{\sqrt {3}}\right)^{2}\left[{\sqrt {3}}+{\sqrt {2}}-\left({\sqrt {3}}-1\right)\left({\sqrt {2}}+1\right){\sqrt {{\sqrt {3}}+{\sqrt {2}}}}\right]^{2}}
λ
∗
(
25
)
=
1
2
(
10
−
2
2
)
(
3
−
2
5
4
)
{\displaystyle \lambda ^{*}(25)={\frac {1}{2}}\left({\sqrt {10}}-2{\sqrt {2}}\right)\left(3-2{\sqrt[{4}]{5}}\right)}
Weitere Lambdafunktionswerte des Schemas λ*(4n - 2) mit n ∈ ℕ können vereinfacht mit dem Tangens dargestellt werden:
λ
∗
(
26
)
=
(
26
+
5
)
(
2
−
1
)
2
tan
[
1
4
π
−
arctan
(
1
3
3
3
+
26
3
−
1
3
3
3
−
26
3
+
1
6
26
−
1
2
2
)
]
4
{\displaystyle \lambda ^{*}(26)=\left({\sqrt {26}}+5\right)\left({\sqrt {2}}-1\right)^{2}\tan \left[{\tfrac {1}{4}}\pi -\arctan \left({\tfrac {1}{3}}{\sqrt[{3}]{3{\sqrt {3}}+{\sqrt {26}}}}-{\tfrac {1}{3}}{\sqrt[{3}]{3{\sqrt {3}}-{\sqrt {26}}}}+{\tfrac {1}{6}}{\sqrt {26}}-{\tfrac {1}{2}}{\sqrt {2}}\right)\right]^{4}}
λ
∗
(
30
)
=
tan
{
1
2
arctan
[
(
10
−
3
)
2
(
5
−
2
)
2
]
}
{\displaystyle \lambda ^{*}(30)=\tan \left\{{\tfrac {1}{2}}\arctan \left[\left({\sqrt {10}}-3\right)^{2}\left({\sqrt {5}}-2\right)^{2}\right]\right\}}
λ
∗
(
34
)
=
tan
{
1
4
arcsin
[
1
9
(
17
−
4
)
2
]
}
{\displaystyle \lambda ^{*}(34)=\tan \left\{{\tfrac {1}{4}}\arcsin \left[{\tfrac {1}{9}}\left({\sqrt {17}}-4\right)^{2}\right]\right\}}
λ
∗
(
42
)
=
tan
{
1
2
arctan
[
(
2
7
−
3
3
)
2
(
2
2
−
7
)
2
]
}
{\displaystyle \lambda ^{*}(42)=\tan \left\{{\tfrac {1}{2}}\arctan \left[\left(2{\sqrt {7}}-3{\sqrt {3}}\right)^{2}\left(2{\sqrt {2}}-{\sqrt {7}}\right)^{2}\right]\right\}}
λ
∗
(
46
)
=
tan
{
1
4
arcsin
[
1
207
(
104
2
−
147
)
]
}
{\displaystyle \lambda ^{*}(46)=\tan \left\{{\tfrac {1}{4}}\arcsin \left[{\tfrac {1}{207}}\left(104{\sqrt {2}}-147\right)\right]\right\}}
λ
∗
(
50
)
=
(
2
−
1
)
tan
[
arctan
(
1
3
5
−
1
3
6
30
+
4
5
3
+
1
3
6
30
−
4
5
3
)
−
1
8
π
]
4
{\displaystyle \lambda ^{*}(50)=\left({\sqrt {2}}-1\right)\tan \left[\arctan \left({\tfrac {1}{3}}{\sqrt {5}}-{\tfrac {1}{3}}{\sqrt[{3}]{6{\sqrt {30}}+4{\sqrt {5}}}}+{\tfrac {1}{3}}{\sqrt[{3}]{6{\sqrt {30}}-4{\sqrt {5}}}}\right)-{\tfrac {1}{8}}\pi \right]^{4}}
λ
∗
(
58
)
=
tan
[
1
4
arcsin
(
1
9801
)
]
{\displaystyle \lambda ^{*}(58)=\tan \left[{\tfrac {1}{4}}\arcsin \left({\tfrac {1}{9801}}\right)\right]}
In jener Liste sind die Lambda-Stern-Werte von Brüchen aufgelistet:
λ
∗
(
1
2
)
=
2
2
−
2
{\displaystyle \lambda ^{*}\left({\frac {1}{2}}\right)={\sqrt {2{\sqrt {2}}-2}}}
λ
∗
(
1
3
)
=
1
4
(
6
+
2
)
{\displaystyle \lambda ^{*}\left({\frac {1}{3}}\right)={\frac {1}{4}}\left({\sqrt {6}}+{\sqrt {2}}\right)}
λ
∗
(
2
3
)
=
(
2
−
3
)
(
3
+
2
)
{\displaystyle \lambda ^{*}\left({\frac {2}{3}}\right)=\left(2-{\sqrt {3}}\right)\left({\sqrt {3}}+{\sqrt {2}}\right)}
λ
∗
(
1
4
)
=
2
2
4
(
2
−
1
)
{\displaystyle \lambda ^{*}\left({\frac {1}{4}}\right)=2{\sqrt[{4}]{2}}\left({\sqrt {2}}-1\right)}
λ
∗
(
3
4
)
=
8
4
(
3
−
2
)
(
2
+
1
)
(
3
−
1
)
3
{\displaystyle \lambda ^{*}\left({\frac {3}{4}}\right)={\sqrt[{4}]{8}}\left({\sqrt {3}}-{\sqrt {2}}\right)\left({\sqrt {2}}+1\right){\sqrt {\left({\sqrt {3}}-1\right)^{3}}}}
λ
∗
(
1
5
)
=
1
2
(
5
−
1
+
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{\displaystyle \lambda ^{*}\left({\frac {1}{5}}\right)={\frac {1}{2}}\left({\sqrt {{\sqrt {5}}-1}}+{\sqrt {3-{\sqrt {5}}}}\right)}
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{\displaystyle \lambda ^{*}\left({\frac {2}{5}}\right)=\left({\sqrt {10}}-3\right)\left({\sqrt {2}}+1\right)^{2}}
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{\displaystyle \lambda ^{*}\left({\frac {3}{5}}\right)={\frac {1}{16}}\left({\sqrt {10}}-{\sqrt {6}}\right)\left(3+{\sqrt {5}}\right)\left(2+{\sqrt {3}}\right)}
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{\displaystyle \lambda ^{*}\left({\frac {4}{5}}\right)=\left({\sqrt {10}}+3\right)\left({\sqrt {5}}+2\right)\left({\sqrt {2}}+1\right)\left({\sqrt {{\sqrt {5}}-1}}-1\right)^{2}}
Die Funktion λ*(x) wird auf folgende Weise[ 5]
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{\displaystyle {\frac {\mathrm {d} }{\mathrm {d} x}}\lambda ^{*}(x)=-{\frac {\lambda ^{*}(x)\lambda ^{*}(1/x)^{2}K[\lambda ^{*}(x)]^{2}}{\pi {\sqrt {x}}}}=-{\frac {\lambda ^{*}(x)\lambda ^{*}(1/x)^{2}K[\lambda ^{*}(x)]^{3}}{\pi K[\lambda ^{*}(1/x)]}}}
Dies wird im nun Folgenden bewiesen. Für die Ableitung des vollständigen elliptischen Integrals erster Art gilt:
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{\displaystyle {\frac {\mathrm {d} }{\mathrm {d} x}}K(x)={\frac {E(x)-(1-x^{2})K(x)}{x(1-x^{2})}}}
Mit der Quotientenregel kann die Umkehrfunktion zur elliptischen Lambda-Stern-Funktion abgeleitet werden:
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{\displaystyle {\frac {\mathrm {d} }{\mathrm {d} x}}{\frac {K\left({\sqrt {1-x^{2}}}\right)^{2}}{K(x)^{2}}}=-{\frac {2K\left({\sqrt {1-x^{2}}}\right)}{x(1-x^{2})K(x)^{3}}}\left[K(x)E\left({\sqrt {1-x^{2}}}\right)+E(x)K\left({\sqrt {1-x^{2}}}\right)-K(x)K\left({\sqrt {1-x^{2}}}\right)\right]}
Die Legendresche Identität [ 6]
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{\displaystyle {\frac {\mathrm {d} }{\mathrm {d} x}}{\frac {K\left({\sqrt {1-x^{2}}}\right)^{2}}{K(x)^{2}}}=-{\frac {\pi K\left({\sqrt {1-x^{2}}}\right)}{x(1-x^{2})K(x)^{3}}}}
Nach der Umkehrregel ist die Ableitung einer Funktion der Kehrwert der Ableitung ihrer Umkehrfunktion mit der Funktion als innere Variable:
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{\displaystyle {\frac {\mathrm {d} }{\mathrm {d} x}}\lambda ^{*}(x)=\left[{\frac {\mathrm {d} }{\mathrm {d} w}}\,{\frac {K\left({\sqrt {1-w^{2}}}\right)^{2}}{K(w)^{2}}}\right]^{-1}[w=\lambda ^{*}(x)]=\left[-{\frac {\pi K\left({\sqrt {1-w^{2}}}\right)}{w(1-w^{2})K(w)^{3}}}\right]^{-1}[w=\lambda ^{*}(x)]=}
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{\displaystyle =\left[-{\frac {w(1-w^{2})K(w)^{3}}{\pi K\left({\sqrt {1-w^{2}}}\right)}}\right][w=\lambda ^{*}(x)]=-{\frac {\lambda ^{*}(x)\lambda ^{*}(1/x)^{2}K[\lambda ^{*}(x)]^{3}}{\pi K[\lambda ^{*}(1/x)]}}}
Chandrasekharan, K. (1985): Elliptic Functions, Grundlehren der mathematischen Wissenschaften . 281, Springer-Verlag, pp. 108–121, ISBN 3-540-15295-4 , Zbl 0575.33001
Reinhardt, W. P.; Walker, P. L. (2010): „Elliptic Modular Function“, in Olver, Frank W. J.; Lozier, Daniel M.; Boisvert, Ronald F.; Clark, Charles W. (eds.), NIST Handbook of Mathematical Functions, Cambridge University Press, ISBN 978-0-521-19225-5 , MR 2723248
Rankin, Robert A. (1977): Modular Forms and Functions . Cambridge University Press, ISBN 0-521-21212-X , Zbl 0376.10020
Jonathan Borwein und Peter Borwein : π and the AGM: A Study in Analytic Number Theory and Computational Complexity . Wiley, 1998, ISBN 978-0-471-31515-5 , Seite 139 (englisch, wiley.com)Milton Abramowitz und Irene Stegun eds. (1972): Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables . New York: Dover Publications, ISBN 978-0-486-61272-0 , Zbl 0543.33001
Nikos Bagis: On the complete solution of the general quintic using the Rogers-Ramanujan continued fraction . Pella, Makedonien, Griechenland, 2015. p. 3, arXiv 1510.00068v1
Folkmar Bornemann, Dirk Laurie, Stan Wagon und Jörg Waldvogel: Vom Lösen numerischer Probleme , Seiten 277 bis 280
↑ complex analysis - Why is the modular $\lambda$ function a quotient of two meromorphic functions in the U.H.P.? Abgerufen am 22. Juli 2021 . ↑ DLMF: 23.15 Definitions. Abgerufen am 22. Juli 2021 . ↑ http://wayback.cecm.sfu.ca/~pborwein/TEMP_PROTECTED/pi-agm.pdf ↑ Eric W. Weisstein: Elliptic Lambda Function. Abgerufen am 22. Juli 2021 (englisch). ↑ Modular lambda function - Fungrim: The Mathematical Functions Grimoire. Abgerufen am 22. Juli 2021 . ↑ integration - Proving Legendres Relation for elliptic curves. Abgerufen am 12. August 2021 . 
![{\displaystyle \lambda (\tau )={\frac {\vartheta _{10}^{4}[\exp(i\pi \tau )]}{\vartheta _{00}^{4}[\exp(i\pi \tau )]}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2e1fbd50eb04506e1add168888ba9c18a681dcef)
![{\displaystyle \vartheta _{10}[\exp(i\pi \tau )]=\sum _{n=-\infty }^{\infty }\exp[i\pi \tau (n+1/2)^{2}]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8a3f645435c91f0f60b7251ef35c993d524ae07d)
![{\displaystyle \vartheta _{00}[\exp(i\pi \tau )]=\sum _{n=-\infty }^{\infty }\exp(i\pi \tau n^{2})}](https://wikimedia.org/api/rest_v1/media/math/render/svg/82ccf56fa4dbbadd5fab70d753f66d5aa0bbcfe7)
![{\displaystyle K\left[{\sqrt {1-\lambda ^{*}(x)^{2}}}\right]/K[\lambda ^{*}(x)]={\sqrt {x}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/521e9a181533cce53feb9316d658be5ae0c6d595)
![{\displaystyle \lambda ^{*}(x)={\frac {\vartheta _{10}^{2}[\exp(-\pi {\sqrt {x}})]}{\vartheta _{00}^{2}[\exp(-\pi {\sqrt {x}})]}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4155c06ea35b7edf4f4b233eb641ee2c030b6e19)
![{\displaystyle \lambda ^{*}(x)={\frac {\vartheta _{01}^{2}[\exp(-\pi /{\sqrt {x}})]}{\vartheta _{00}^{2}[\exp(-\pi /{\sqrt {x}})]}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/307c8de7f602fd97642df6cc7418ab6dc3f5e1ce)
![{\displaystyle \lambda ^{*}(x)={\frac {\vartheta _{10}\left[{\tfrac {1}{4}}\pi ;\exp \left(-{\tfrac {1}{2}}\pi {\sqrt {x}}\right)\right]^{4}}{\vartheta _{00}\left[{\tfrac {1}{4}}\pi ;\exp \left(-{\tfrac {1}{2}}\pi {\sqrt {x}}\right)\right]^{4}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b63b0c8ebaa399a175d61ab9aa2d8944f4bf51a6)
![{\displaystyle \vartheta _{00}(v;w)=\prod _{n=1}^{\infty }\left(1-w^{2n}\right)\left[1+2\cos(2v)w^{2n-1}+w^{4n-2}\right]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/bedb4625c7e74f359a54442308cf62684590fd64)
![{\displaystyle \vartheta _{01}(v;w)=\prod _{n=1}^{\infty }\left(1-w^{2n}\right)\left[1-2\cos(2v)w^{2n-1}+w^{4n-2}\right]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ead3822cb0a3bec4900ab2b5e51f47317260e5c4)
![{\displaystyle \vartheta _{10}(v;w)=2w^{1/4}\cos(v)\prod _{n=1}^{\infty }\left(1-w^{2n}\right)\left[1+2\cos(2v)w^{2n}+w^{4n}\right]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ac814488bd8696feb7978f186f2ab7898a833adc)
![{\displaystyle \lambda ^{*}(x)={\biggl \{}\sum _{a=-\infty }^{\infty }\exp {\bigl [}-(a+1/2)^{2}\pi {\sqrt {x}}{\bigr ]}{\biggr \}}^{2}{\biggl \{}\sum _{a=-\infty }^{\infty }\exp(-a^{2}\pi {\sqrt {x}}){\biggr \}}^{-2}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/77bd779b6c056f1230136661d4f37e9d4a0f7ed0)
![{\displaystyle \lambda ^{*}(x)={\biggl \{}\sum _{a=-\infty }^{\infty }\operatorname {sech} {\bigl [}(a+1/2)\pi {\sqrt {x}}{\bigr ]}{\biggr \}}{\biggl \{}\sum _{a=-\infty }^{\infty }\operatorname {sech} (a\pi {\sqrt {x}}){\biggr \}}^{-1}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f2935aaf660d984f86e5a08a8656e0a78ce4abe0)
![{\displaystyle \lambda ^{*}(x)={\biggl [}\sum _{a=-\infty }^{\infty }(-1)^{a}\exp {\biggl (}-{\frac {a^{2}\pi }{\sqrt {x}}}{\biggr )}{\biggr ]}^{2}{\biggl [}\sum _{a=-\infty }^{\infty }\exp \left(-{\frac {a^{2}\pi }{\sqrt {x}}}\right){\biggr ]}^{-2}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/86ce830a6ebaf72834ce1d6f0bb2776291a30a3e)
![{\displaystyle \lambda ^{*}(x)=\prod _{a=0}^{\infty }\operatorname {tanh} {\biggl [}{\frac {(a+1/2)\pi }{\sqrt {x}}}{\biggr ]}^{4}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e103e4a22a6babd89049eaf7ec5cd7707d3977a9)
![{\displaystyle \vartheta _{00}[\exp(-\pi {\sqrt {x}})]=1+2\exp(-\pi {\sqrt {x}})\int _{0}^{\infty }\exp \left(-\pi y^{2}\right){\frac {\exp(2\pi {\sqrt {x}})-\cos(2\pi {\sqrt[{4}]{x}}\,y)}{\cosh(2\pi {\sqrt {x}})-\cos(2\pi {\sqrt[{4}]{x}}\,y)}}\mathrm {d} y}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9623cb728a9490af1e60d4fa0d5cf4a8fe0b9ae4)
![{\displaystyle \vartheta _{00}[\exp(-2\pi {\sqrt {x}})]=1+2\exp(-2\pi {\sqrt {x}})\int _{0}^{\infty }\exp \left(-\pi y^{2}\right){\frac {\exp(4\pi {\sqrt {x}})-\cos(2\pi {\sqrt[{4}]{4x}}\,y)}{\cosh(4\pi {\sqrt {x}})-\cos(2\pi {\sqrt[{4}]{4x}}\,y)}}\mathrm {d} y}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7e18bba418e3618eff499a4206814dbaddcffead)
![{\displaystyle \lambda ^{*}(x)=\sin {\biggl \langle }2\arccos {\biggl \{}{\frac {\vartheta _{00}[\exp(-2\pi {\sqrt {x}})]}{\vartheta _{00}[\exp(-\pi {\sqrt {x}})]}}{\biggr \}}{\biggr \rangle }}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d0ad33fd26018f6f97eb11d98d31c918a8bc9a32)
![{\displaystyle \vartheta _{10}[\exp(-\pi {\sqrt {x}})]=\sum _{a=-\infty }^{\infty }\exp \left[-\left(a+{\tfrac {1}{2}}\right)^{2}\pi {\sqrt {x}}\right]={\biggl \{}\sum _{a=-\infty }^{\infty }\operatorname {sech} \left[\left(a+{\tfrac {1}{2}}\right)\pi {\sqrt {x}}\right]{\biggr \}}^{1/2}=}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4dbbad42c2bd46d1cd78539985369dd9eb1a5f96)
![{\displaystyle ={\sqrt {2\pi ^{-1}\lambda ^{*}(x)K[\lambda ^{*}(x)]}}={\sqrt[{4}]{\lambda ^{*}(4x)}}{\sqrt {4\pi ^{-1}K[\lambda ^{*}(4x)]}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/bb0f7dee10a347aea4fbfee6a8913f8aa8ff09b5)
![{\displaystyle \vartheta _{00}[\exp(-\pi {\sqrt {x}})]=\sum _{a=-\infty }^{\infty }\exp(-a^{2}\pi {\sqrt {x}})={\biggl [}\sum _{a=-\infty }^{\infty }\operatorname {sech} (a\pi {\sqrt {x}}){\biggr ]}^{1/2}={\sqrt {2\pi ^{-1}K[\lambda ^{*}(x)]}}=\operatorname {agm} [1;\lambda ^{*}(1/x)]^{-1/2}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5998c996032f6c10d271a79ba0bd3e9b68b226dc)
![{\displaystyle \vartheta _{01}[\exp(-\pi {\sqrt {x}})]=\sum _{a=-\infty }^{\infty }(-1)^{a}\exp(-a^{2}\pi {\sqrt {x}})={\sqrt {2\pi ^{-1}\lambda ^{*}(1/x)K[\lambda ^{*}(x)]}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0dc37a9c0e0989a445915897c113450991e15460)
![{\displaystyle \sum _{n=1}^{\infty }{\frac {1}{F_{2n-1}}}=\sum _{n=1}^{\infty }{\frac {\sqrt {5}}{2}}\operatorname {sech} \left[\left(n-{\tfrac {1}{2}}\right)\operatorname {arcosh} \left({\tfrac {3}{2}}\right)\right]={\frac {\sqrt {5}}{4}}\sum _{a=-\infty }^{\infty }\operatorname {sech} \left[\left(a+{\tfrac {1}{2}}\right)\operatorname {arcosh} \left({\tfrac {3}{2}}\right)\right]=}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e56f4dae6d2614045a3f68cc4a92e9e9db3b1e2f)
![{\displaystyle ={\frac {\sqrt {5}}{4}}\vartheta _{10}(\Phi ^{-2})^{2}={\frac {\sqrt {5}}{8}}{\bigl [}\vartheta _{00}(\Phi ^{-1})^{2}-\vartheta _{01}(\Phi ^{-1})^{2}{\bigr ]}={\frac {\sqrt {5}}{\pi }}{\sqrt {\lambda ^{*}[16\,\pi ^{-2}\ln(\Phi )^{2}]}}K\{\lambda ^{*}[16\,\pi ^{-2}\ln(\Phi )^{2}]\}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/519961e7831c6ca87146e8344fe73681b17248f3)
![{\displaystyle \sum _{n=1}^{\infty }{\frac {1}{P(2n-1)}}={\frac {2{\sqrt {2}}}{\pi }}{\sqrt {\lambda ^{*}[16\,\pi ^{-2}\operatorname {arsinh} (1)^{2}]}}K\{\lambda ^{*}[16\,\pi ^{-2}\operatorname {arsinh} (1)^{2}]\}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a06feaaf5813cac11eda47f6e9b393d1f2ac0f13)
![{\displaystyle {\sqrt {n}}=\sum _{a=1}^{n}\operatorname {dn} {\biggl \{}{\frac {2a}{n}}K{\biggl [}\lambda ^{*}{\biggl (}{\frac {1}{n}}{\biggr )}{\biggr ]};\lambda ^{*}{\biggl (}{\frac {1}{n}}{\biggr )}{\biggr \}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f49364c1637e5f2e7dd47d34507f5472ea1ce20a)
![{\displaystyle \lambda ^{*}(n^{2}x)=\lambda ^{*}(x)^{n}\prod _{a=1}^{n}\operatorname {sn} {\biggl \{}{\frac {2a-1}{n}}K{\bigl [}\lambda ^{*}(x){\bigr ]};\lambda ^{*}(x){\biggr \}}^{2}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/91022ae7e3fa4e645d65c1d12ab683c25a307fd1)
![{\displaystyle [\lambda ^{*}(x)+1][\lambda ^{*}(4/x)+1]=2}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7e015a4b8dbb80bb4bece1c0c5224457cda46f58)
![{\displaystyle \lambda ^{*}(4x)={\frac {1-{\sqrt {1-\lambda ^{*}(x)^{2}}}}{1+{\sqrt {1-\lambda ^{*}(x)^{2}}}}}={\frac {\lambda ^{*}(x)^{2}}{{\bigl [}\,1+{\sqrt {1-\lambda ^{*}(x)^{2}}}\,{\bigr ]}^{2}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4a555522a7927ade82463d274220ddff50124f04)
![{\displaystyle \lambda ^{*}(4x)=\tan \left\{{\tfrac {1}{2}}\arcsin[\lambda ^{*}(x)]\right\}^{2}=\tanh \left\{{\tfrac {1}{2}}\operatorname {artanh} [\lambda ^{*}(x)]\right\}^{2}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/430ac8a0567dc4108fce8810cbaad7ede07be578)
![{\displaystyle [\lambda ^{*}(x)-\lambda ^{*}(9x)]^{4}=16\lambda ^{*}(x)\lambda ^{*}(9x)\left[1-\lambda ^{*}(x)^{2}\right]\left[1-\lambda ^{*}(9x)^{2}\right]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8d58ee8b7c4a919ae73ca2ac7560dec21f6e38fd)
![{\displaystyle \lambda ^{*}(x)-\lambda ^{*}(9x)=2[\lambda ^{*}(x)\lambda ^{*}(9x)]^{1/4}-2[\lambda ^{*}(x)\lambda ^{*}(9x)]^{3/4}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/de8a0866508ba81524c948c3c4f6cf7ab4f23c0d)
![{\displaystyle {\sqrt {\lambda ^{*}(x)\lambda ^{*}(9x)}}+{\sqrt[{4}]{\left[1-\lambda ^{*}(x)^{2}\right]\left[1-\lambda ^{*}(9x)^{2}\right]}}=1}](https://wikimedia.org/api/rest_v1/media/math/render/svg/519b66c45d1a8e2e6dbd9c76f77601bbd03d2897)
![{\displaystyle {\biggl [}{\frac {2\lambda ^{*}(x)}{1-\lambda ^{*}(x)^{2}}}{\biggr ]}^{1/2}-{\biggl [}{\frac {2\lambda ^{*}(25x)}{1-\lambda ^{*}(25x)^{2}}}{\biggr ]}^{1/2}=2{\biggl [}{\frac {2\lambda ^{*}(x)}{1-\lambda ^{*}(x)^{2}}}{\biggr ]}^{1/12}{\biggl [}{\frac {2\lambda ^{*}(25x)}{1-\lambda ^{*}(25x)^{2}}}{\biggr ]}^{1/12}+2{\biggl [}{\frac {2\lambda ^{*}(x)}{1-\lambda ^{*}(x)^{2}}}{\biggr ]}^{5/12}{\biggl [}{\frac {2\lambda ^{*}(25x)}{1-\lambda ^{*}(25x)^{2}}}{\biggr ]}^{5/12}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0b9d7602d8782f5df72d52af013c757817f50e65)
![{\displaystyle \left[\lambda ^{*}(x)^{1/2}-\lambda ^{*}(25x)^{1/2}\right]\left[\lambda ^{*}(x)+\lambda ^{*}(25x)+6\lambda ^{*}(x)^{1/2}\lambda ^{*}(25x)^{1/2}\right]=4\lambda ^{*}(x)^{1/4}\lambda ^{*}(25x)^{1/4}[1-\lambda ^{*}(x)\lambda ^{*}(25x)]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/496fddb0cf0429130e8d5deea4538817e0b8c7f3)
![{\displaystyle \lambda ^{*}(x)\lambda ^{*}(25x)+{\sqrt {[1-\lambda ^{*}(x)]^{2}[1-\lambda ^{*}(25x)]^{2}}}+2{\sqrt[{6}]{16\lambda ^{*}(x)^{2}\lambda ^{*}(25x)^{2}\left[1-\lambda ^{*}(x)^{2}\right]\left[1-\lambda ^{*}(25x)^{2}\right]}}=1}](https://wikimedia.org/api/rest_v1/media/math/render/svg/44bc31994a604545188229ec013a0c4f19cd5a7a)
![{\displaystyle {\sqrt[{4}]{\lambda ^{*}(x)\lambda ^{*}(49x)}}+{\sqrt[{8}]{\left[1-\lambda ^{*}(x)^{2}\right]\left[1-\lambda ^{*}(49x)^{2}\right]}}=1}](https://wikimedia.org/api/rest_v1/media/math/render/svg/03f2c3f65a3c3fd7982a6c2d751d8c8ffef86d9a)
![{\displaystyle {\sqrt {\lambda ^{*}(x)\lambda ^{*}(121x)}}+{\sqrt[{4}]{\left[1-\lambda ^{*}(x)^{2}\right]\left[1-\lambda ^{*}(121x)^{2}\right]}}+2{\sqrt[{12}]{16\lambda ^{*}(x)^{2}\lambda ^{*}(121x)^{2}\left[1-\lambda ^{*}(x)^{2}\right]\left[1-\lambda ^{*}(121x)^{2}\right]}}=1}](https://wikimedia.org/api/rest_v1/media/math/render/svg/bead3db040dd3d1ab0eb1495ad9cb6ee60403499)
![{\displaystyle {\sqrt[{4}]{\lambda ^{*}(x)\lambda ^{*}(529x)}}+{\sqrt[{8}]{\left[1-\lambda ^{*}(x)^{2}\right]\left[1-\lambda ^{*}(529x)^{2}\right]}}+{\sqrt {2}}{\sqrt[{24}]{16\lambda ^{*}(x)^{2}\lambda ^{*}(529x)^{2}\left[1-\lambda ^{*}(x)^{2}\right]\left[1-\lambda ^{*}(529x)^{2}\right]}}=1}](https://wikimedia.org/api/rest_v1/media/math/render/svg/77e886d62813a61c17a22fbd44db27d660d4b45c)
![{\displaystyle G(x)=\sin\{2\arcsin[\lambda ^{*}(x)]\}^{-1/12}=1\left/\left[{\sqrt[{12}]{2\lambda ^{*}(x)}}{\sqrt[{24}]{1-\lambda ^{*}(x)^{2}}}\right]\right.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7ccdc6b11c839ec51f0b91c8f00edbd0bcb2c56a)
![{\displaystyle g(x)=\tan\{2\arctan[\lambda ^{*}(x)]\}^{-1/12}={\sqrt[{12}]{[1-\lambda ^{*}(x)^{2}]/[2\lambda ^{*}(x)]}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0faafcdf649c59a5287a89cee5bf869a61f19229)
![{\displaystyle \lambda ^{*}(x)=\tan \left\{{\tfrac {1}{2}}\arctan \left[g(x)^{-12}\right]\right\}={\sqrt {g(x)^{24}+1}}-g(x)^{12}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/882c3dc2f43ab8e419576d72506e6c545b2d3828)
![{\displaystyle \lambda ^{*}(9)={\frac {1}{2}}\left({\sqrt {3}}-1\right)\left({\sqrt {2}}-{\sqrt[{4}]{3}}\right)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d1bfce605ad029509578b8837da4d02baa92df95)
![{\displaystyle \lambda ^{*}(11)={\frac {1}{16}}\left({\sqrt {22}}+3{\sqrt {2}}\right)\left({\frac {1}{3}}{\sqrt[{3}]{6{\sqrt {3}}+2{\sqrt {11}}}}-{\frac {1}{3}}{\sqrt[{3}]{6{\sqrt {3}}-2{\sqrt {11}}}}+{\frac {1}{3}}{\sqrt {11}}-1\right)^{4}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/02571366e2d6d7e2882b7b83f738943a31536e6a)
![{\displaystyle \lambda ^{*}(16)=\left({\sqrt {2}}+1\right)^{2}\left({\sqrt[{4}]{2}}-1\right)^{4}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/06cfdd192b23e6019daa9381c648e6da1ecefb8c)
![{\displaystyle \lambda ^{*}(17)={\frac {1}{4}}\left({\sqrt {7+{\sqrt {17}}}}-{\sqrt {{\sqrt {17}}+3}}\right)\left({\sqrt {5+{\sqrt {17}}}}-{\sqrt[{4}]{10{\sqrt {17}}+38}}\right)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d435562306fb91c30f1c18e25ad4008c8c05c31c)
![{\displaystyle \lambda ^{*}(19)={\frac {1}{16}}\left(3{\sqrt {38}}+13{\sqrt {2}}\right)\left[{\frac {1}{6}}\left({\sqrt {19}}-2+{\sqrt {3}}\right){\sqrt[{3}]{3{\sqrt {3}}-{\sqrt {19}}}}-{\frac {1}{6}}\left({\sqrt {19}}-2-{\sqrt {3}}\right){\sqrt[{3}]{3{\sqrt {3}}+{\sqrt {19}}}}-{\frac {1}{3}}\left(5-{\sqrt {19}}\right)\right]^{4}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5345de3430c61ca8b9a4fe1f0bc53d23d7161a10)
![{\displaystyle \lambda ^{*}(21)={\frac {1}{4}}\left[\left({\sqrt {7}}-{\sqrt {3}}\right)\left({\sqrt {3}}+1\right){\sqrt {{\sqrt {7}}-2}}-2{\sqrt {14}}+5{\sqrt {2}}-{\sqrt {42}}+2{\sqrt {6}}\right]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3d57d3f65e4518b9b1b65a7e1a16db7ae09ed58f)
![{\displaystyle \lambda ^{*}(23)={\frac {1}{32}}\left(5{\sqrt {2}}+{\sqrt {46}}\right)\left[{\frac {2}{3}}+{\frac {1}{6}}\left({\sqrt {3}}+1\right){\sqrt[{3}]{100-12{\sqrt {69}}}}-{\frac {1}{6}}\left({\sqrt {3}}-1\right){\sqrt[{3}]{100+12{\sqrt {69}}}}\right]^{4}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0ee2858de9eb95df8e1112af10a8cd49e58f26b2)
![{\displaystyle \lambda ^{*}(24)=\left(2+{\sqrt {3}}\right)^{2}\left[{\sqrt {3}}+{\sqrt {2}}-\left({\sqrt {3}}-1\right)\left({\sqrt {2}}+1\right){\sqrt {{\sqrt {3}}+{\sqrt {2}}}}\right]^{2}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/fae0041ca32abf9d6744f9d0d1c3220aff27fddc)
![{\displaystyle \lambda ^{*}(25)={\frac {1}{2}}\left({\sqrt {10}}-2{\sqrt {2}}\right)\left(3-2{\sqrt[{4}]{5}}\right)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ae051f5f5d6e8ffced13c2ee8a6b01449b8035a8)
![{\displaystyle \lambda ^{*}(26)=\left({\sqrt {26}}+5\right)\left({\sqrt {2}}-1\right)^{2}\tan \left[{\tfrac {1}{4}}\pi -\arctan \left({\tfrac {1}{3}}{\sqrt[{3}]{3{\sqrt {3}}+{\sqrt {26}}}}-{\tfrac {1}{3}}{\sqrt[{3}]{3{\sqrt {3}}-{\sqrt {26}}}}+{\tfrac {1}{6}}{\sqrt {26}}-{\tfrac {1}{2}}{\sqrt {2}}\right)\right]^{4}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c710fbd5a77591f40e6f090f9213679ac416a163)
![{\displaystyle \lambda ^{*}(30)=\tan \left\{{\tfrac {1}{2}}\arctan \left[\left({\sqrt {10}}-3\right)^{2}\left({\sqrt {5}}-2\right)^{2}\right]\right\}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/50bfa27cb6a42f74b9a8f5df091925196b31efec)
![{\displaystyle \lambda ^{*}(34)=\tan \left\{{\tfrac {1}{4}}\arcsin \left[{\tfrac {1}{9}}\left({\sqrt {17}}-4\right)^{2}\right]\right\}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d1546d608fd7fd30879765e8d3caf833e7e4d7e8)
![{\displaystyle \lambda ^{*}(42)=\tan \left\{{\tfrac {1}{2}}\arctan \left[\left(2{\sqrt {7}}-3{\sqrt {3}}\right)^{2}\left(2{\sqrt {2}}-{\sqrt {7}}\right)^{2}\right]\right\}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/efb1ebe9152ecac72a78aab754c2ef8bb4a0ffb4)
![{\displaystyle \lambda ^{*}(46)=\tan \left\{{\tfrac {1}{4}}\arcsin \left[{\tfrac {1}{207}}\left(104{\sqrt {2}}-147\right)\right]\right\}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8c8d61e0720962c8b0a787506dba702e29d130cd)
![{\displaystyle \lambda ^{*}(50)=\left({\sqrt {2}}-1\right)\tan \left[\arctan \left({\tfrac {1}{3}}{\sqrt {5}}-{\tfrac {1}{3}}{\sqrt[{3}]{6{\sqrt {30}}+4{\sqrt {5}}}}+{\tfrac {1}{3}}{\sqrt[{3}]{6{\sqrt {30}}-4{\sqrt {5}}}}\right)-{\tfrac {1}{8}}\pi \right]^{4}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/dc298b4ae6841be1a815ffc7b2b62f6c38cecc69)
![{\displaystyle \lambda ^{*}(58)=\tan \left[{\tfrac {1}{4}}\arcsin \left({\tfrac {1}{9801}}\right)\right]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ca0babc1cd5b68da6c87ab260182a94de4d152a5)
![{\displaystyle \lambda ^{*}\left({\frac {1}{4}}\right)=2{\sqrt[{4}]{2}}\left({\sqrt {2}}-1\right)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7af0f187e95969ba49957c5ade281a35a3a9c6e8)
![{\displaystyle \lambda ^{*}\left({\frac {3}{4}}\right)={\sqrt[{4}]{8}}\left({\sqrt {3}}-{\sqrt {2}}\right)\left({\sqrt {2}}+1\right){\sqrt {\left({\sqrt {3}}-1\right)^{3}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f00ac1dcf9ce55f31896a7f191c768954b02497c)
![{\displaystyle {\frac {\mathrm {d} }{\mathrm {d} x}}\lambda ^{*}(x)=-{\frac {\lambda ^{*}(x)\lambda ^{*}(1/x)^{2}K[\lambda ^{*}(x)]^{2}}{\pi {\sqrt {x}}}}=-{\frac {\lambda ^{*}(x)\lambda ^{*}(1/x)^{2}K[\lambda ^{*}(x)]^{3}}{\pi K[\lambda ^{*}(1/x)]}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f8b90067a9ca00dcc538abd3ca0274689bb7631e)
![{\displaystyle {\frac {\mathrm {d} }{\mathrm {d} x}}{\frac {K\left({\sqrt {1-x^{2}}}\right)^{2}}{K(x)^{2}}}=-{\frac {2K\left({\sqrt {1-x^{2}}}\right)}{x(1-x^{2})K(x)^{3}}}\left[K(x)E\left({\sqrt {1-x^{2}}}\right)+E(x)K\left({\sqrt {1-x^{2}}}\right)-K(x)K\left({\sqrt {1-x^{2}}}\right)\right]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8f96666327094abe0f7b9a61bf9920dddc11d93e)
![{\displaystyle {\frac {\mathrm {d} }{\mathrm {d} x}}\lambda ^{*}(x)=\left[{\frac {\mathrm {d} }{\mathrm {d} w}}\,{\frac {K\left({\sqrt {1-w^{2}}}\right)^{2}}{K(w)^{2}}}\right]^{-1}[w=\lambda ^{*}(x)]=\left[-{\frac {\pi K\left({\sqrt {1-w^{2}}}\right)}{w(1-w^{2})K(w)^{3}}}\right]^{-1}[w=\lambda ^{*}(x)]=}](https://wikimedia.org/api/rest_v1/media/math/render/svg/204f13587e2affa76881af70d403052409eaf8d1)
![{\displaystyle =\left[-{\frac {w(1-w^{2})K(w)^{3}}{\pi K\left({\sqrt {1-w^{2}}}\right)}}\right][w=\lambda ^{*}(x)]=-{\frac {\lambda ^{*}(x)\lambda ^{*}(1/x)^{2}K[\lambda ^{*}(x)]^{3}}{\pi K[\lambda ^{*}(1/x)]}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/fcabffa0c425b6bf887ff132a555eb5638c80d17)