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.
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):=\{{\begin{pmatrix}a&b\\c&d\end{pmatrix}}\in \operatorname {SL} _{2}(\mathbb {Z} )|a\equiv d\equiv 1(\operatorname {mod} 2),b\equiv c\equiv 0(\operatorname {mod} 2)\}=\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}}\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 )}}}
Definition von Lambda-Stern
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[{\sqrt {1-\lambda ^{*}(x)^{2}}}]/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}[{\tfrac {1}{4}}\pi ;\exp(-{\tfrac {1}{2}}\pi {\sqrt {x}})]^{4}}{\vartheta _{00}[{\tfrac {1}{4}}\pi ;\exp(-{\tfrac {1}{2}}\pi {\sqrt {x}})]^{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 }(1-w^{2n})[1+2\cos(2v)w^{2n-1}+w^{4n-2}]}
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 }(1-w^{2n})[1-2\cos(2v)w^{2n-1}+w^{4n-2}]}
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 }(1-w^{2n})[1+2\cos(2v)w^{2n}+w^{4n}]}
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)}
Definitionen als Summenreihen und Produktreihen
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}}
Definition mit Integralen
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(-\pi y^{2}){\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(-\pi y^{2}){\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 }}
Eigenschaften
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 \{\lambda ,{\frac {1}{1-\lambda }},{\frac {\lambda -1}{\lambda }},{\frac {1}{\lambda }},{\frac {\lambda }{\lambda -1}},1-\lambda \}}
Algebraische Beziehungen von Lambda-Stern
Spezielle Werte
Ableitung
Die Funktion λ*(x) wird auf folgende Weise[ 5]
d
d
x
λ
∗
(
x
)
=
−
λ
∗
(
x
)
λ
∗
(
1
/
x
)
2
K
[
λ
∗
(
x
)
]
2
π
x
=
−
λ
∗
(
x
)
λ
∗
(
1
/
x
)
2
K
[
λ
∗
(
x
)
]
3
π
K
[
λ
∗
(
1
/
x
)
]
{\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:
d
d
x
K
(
x
)
=
E
(
x
)
−
(
1
−
x
2
)
K
(
x
)
x
(
1
−
x
2
)
{\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:
d
d
x
K
(
1
−
x
2
)
2
K
(
x
)
2
=
−
2
K
(
1
−
x
2
)
x
(
1
−
x
2
)
K
(
x
)
3
[
K
(
x
)
E
(
1
−
x
2
)
+
E
(
x
)
K
(
1
−
x
2
)
−
K
(
x
)
K
(
1
−
x
2
)
]
{\displaystyle {\frac {\mathrm {d} }{\mathrm {d} x}}{\frac {K({\sqrt {1-x^{2}}})^{2}}{K(x)^{2}}}=-{\frac {2K({\sqrt {1-x^{2}}})}{x(1-x^{2})K(x)^{3}}}{\bigl [}K(x)E({\sqrt {1-x^{2}}})+E(x)K({\sqrt {1-x^{2}}})-K(x)K({\sqrt {1-x^{2}}}){\bigr ]}}
Die Legendresche Identität [ 6]
d
d
x
K
(
1
−
x
2
)
2
K
(
x
)
2
=
−
π
K
(
1
−
x
2
)
x
(
1
−
x
2
)
K
(
x
)
3
{\displaystyle {\frac {\mathrm {d} }{\mathrm {d} x}}{\frac {K({\sqrt {1-x^{2}}})^{2}}{K(x)^{2}}}=-{\frac {\pi K({\sqrt {1-x^{2}}})}{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:
d
d
x
λ
∗
(
x
)
=
[
d
d
w
K
(
1
−
w
2
)
2
K
(
w
)
2
]
−
1
[
w
=
λ
∗
(
x
)
]
=
[
−
π
K
(
1
−
w
2
)
w
(
1
−
w
2
)
K
(
w
)
3
]
−
1
[
w
=
λ
∗
(
x
)
]
=
{\displaystyle {\frac {\mathrm {d} }{\mathrm {d} x}}\lambda ^{*}(x)={\biggl [}{\frac {\mathrm {d} }{\mathrm {d} w}}\,{\frac {K({\sqrt {1-w^{2}}})^{2}}{K(w)^{2}}}{\biggr ]}^{-1}[w=\lambda ^{*}(x)]={\biggl [}-{\frac {\pi K({\sqrt {1-w^{2}}})}{w(1-w^{2})K(w)^{3}}}{\biggr ]}^{-1}[w=\lambda ^{*}(x)]=}
=
[
−
w
(
1
−
w
2
)
K
(
w
)
3
π
K
(
1
−
w
2
)
]
[
w
=
λ
∗
(
x
)
]
=
−
λ
∗
(
x
)
λ
∗
(
1
/
x
)
2
K
[
λ
∗
(
x
)
]
3
π
K
[
λ
∗
(
1
/
x
)
]
{\displaystyle ={\biggl [}-{\frac {w(1-w^{2})K(w)^{3}}{\pi K({\sqrt {1-w^{2}}})}}{\biggr ]}[w=\lambda ^{*}(x)]=-{\frac {\lambda ^{*}(x)\lambda ^{*}(1/x)^{2}K[\lambda ^{*}(x)]^{3}}{\pi K[\lambda ^{*}(1/x)]}}}
Literatur
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
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
Weblinks
Einzelnachweise 
![{\displaystyle \lambda (\tau )={\frac {\vartheta _{10}^{4}[\exp(i\pi \tau )]}{\vartheta _{00}^{4}[\exp(i\pi \tau )]}}}](/media/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}]}](/media/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})}](/media/api/rest_v1/media/math/render/svg/82ccf56fa4dbbadd5fab70d753f66d5aa0bbcfe7)
![{\displaystyle K[{\sqrt {1-\lambda ^{*}(x)^{2}}}]/K[\lambda ^{*}(x)]={\sqrt {x}}}](/media/api/rest_v1/media/math/render/svg/40f704d34404fc3d81e644cc5691f8a83216d5b4)
![{\displaystyle \lambda ^{*}(x)={\frac {\vartheta _{10}^{2}[\exp(-\pi {\sqrt {x}})]}{\vartheta _{00}^{2}[\exp(-\pi {\sqrt {x}})]}}}](/media/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}})]}}}](/media/api/rest_v1/media/math/render/svg/307c8de7f602fd97642df6cc7418ab6dc3f5e1ce)
![{\displaystyle \lambda ^{*}(x)={\frac {\vartheta _{10}[{\tfrac {1}{4}}\pi ;\exp(-{\tfrac {1}{2}}\pi {\sqrt {x}})]^{4}}{\vartheta _{00}[{\tfrac {1}{4}}\pi ;\exp(-{\tfrac {1}{2}}\pi {\sqrt {x}})]^{4}}}}](/media/api/rest_v1/media/math/render/svg/c7d24f3dcf5b589841c13bee744249026d1f06fe)
![{\displaystyle \vartheta _{00}(v;w)=\prod _{n=1}^{\infty }(1-w^{2n})[1+2\cos(2v)w^{2n-1}+w^{4n-2}]}](/media/api/rest_v1/media/math/render/svg/2c5acc090a4c8a2327974fa27125ca0f796ce535)
![{\displaystyle \vartheta _{01}(v;w)=\prod _{n=1}^{\infty }(1-w^{2n})[1-2\cos(2v)w^{2n-1}+w^{4n-2}]}](/media/api/rest_v1/media/math/render/svg/05ceef12dd801305a4cf783265e8e157f4bdffde)
![{\displaystyle \vartheta _{10}(v;w)=2w^{1/4}\cos(v)\prod _{n=1}^{\infty }(1-w^{2n})[1+2\cos(2v)w^{2n}+w^{4n}]}](/media/api/rest_v1/media/math/render/svg/89a19ed41533c1761bf80ff54c7445c2518b5fe6)
![{\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}}](/media/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}}](/media/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}}](/media/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}}](/media/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(-\pi y^{2}){\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}](/media/api/rest_v1/media/math/render/svg/e224bd26f7ad23a782ff9162a9c2d3c429f54b27)
![{\displaystyle \vartheta _{00}[\exp(-2\pi {\sqrt {x}})]=1+2\exp(-2\pi {\sqrt {x}})\int _{0}^{\infty }\exp(-\pi y^{2}){\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}](/media/api/rest_v1/media/math/render/svg/ba2de8b35d771548ca1053d0f827ccbf42f3ae2c)
![{\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 }}](/media/api/rest_v1/media/math/render/svg/d0ad33fd26018f6f97eb11d98d31c918a8bc9a32)
![{\displaystyle {\sqrt {n}}=\sum _{a=1}^{n}{\text{dn}}{\biggl \{}{\frac {2a}{n}}K{\biggl [}\lambda ^{*}{\biggl (}{\frac {1}{n}}{\biggr )}{\biggr ]};\lambda ^{*}{\biggl (}{\frac {1}{n}}{\biggr )}{\biggr \}}}](/media/api/rest_v1/media/math/render/svg/53bd5625e50f49309c3e67f591a4e718ba0dbc64)
![{\displaystyle \lambda ^{*}(n^{2}x)=\lambda ^{*}(x)^{n}\prod _{a=1}^{n}{\text{sn}}{\biggl \{}{\frac {2a-1}{n}}K{\bigl [}\lambda ^{*}(x){\bigr ]};\lambda ^{*}(x){\biggr \}}^{2}}](/media/api/rest_v1/media/math/render/svg/a44df71ed42096236726d9ad6203f64bbd32381a)
![{\displaystyle [\lambda ^{*}(x)+1][\lambda ^{*}(4/x)+1]=2}](/media/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}}}}](/media/api/rest_v1/media/math/render/svg/4a555522a7927ade82463d274220ddff50124f04)
![{\displaystyle \lambda ^{*}(4x)=\tan\{{\tfrac {1}{2}}\arcsin[\lambda ^{*}(x)]\}^{2}=\tanh\{{\tfrac {1}{2}}\operatorname {artanh} [\lambda ^{*}(x)]\}^{2}}](/media/api/rest_v1/media/math/render/svg/181f3609c93f07555cd839796d9732b7f0f56c89)
![{\displaystyle [\lambda ^{*}(x)-\lambda ^{*}(9x)]^{4}=16\lambda ^{*}(x)\lambda ^{*}(9x)[1-\lambda ^{*}(x)^{2}][1-\lambda ^{*}(9x)^{2}]}](/media/api/rest_v1/media/math/render/svg/032fc25d5882b90d1a859af87b54f7fb00ecc59a)
![{\displaystyle \lambda ^{*}(x)-\lambda ^{*}(9x)=2[\lambda ^{*}(x)\lambda ^{*}(9x)]^{1/4}-2[\lambda ^{*}(x)\lambda ^{*}(9x)]^{3/4}}](/media/api/rest_v1/media/math/render/svg/de8a0866508ba81524c948c3c4f6cf7ab4f23c0d)
![{\displaystyle {\sqrt {\lambda ^{*}(x)\lambda ^{*}(9x)}}+{\sqrt[{4}]{[1-\lambda ^{*}(x)^{2}][1-\lambda ^{*}(9x)^{2}]}}=1}](/media/api/rest_v1/media/math/render/svg/4f0256fda0729a1c3bdb4734274bc2035d6417fb)
![{\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}}](/media/api/rest_v1/media/math/render/svg/0b9d7602d8782f5df72d52af013c757817f50e65)
![{\displaystyle [\lambda ^{*}(x)^{1/2}-\lambda ^{*}(25x)^{1/2}][\lambda ^{*}(x)+\lambda ^{*}(25x)+6\lambda ^{*}(x)^{1/2}\lambda ^{*}(25x)^{1/2}]=4\lambda ^{*}(x)^{1/4}\lambda ^{*}(25x)^{1/4}[1-\lambda ^{*}(x)\lambda ^{*}(25x)]}](/media/api/rest_v1/media/math/render/svg/3e75f1cbf1304027d48797b38a8629bdb2ed3c46)
![{\displaystyle \lambda ^{*}(x)\lambda ^{*}(25x)+{\sqrt {[1-\lambda ^{*}(x)]^{2}[1-\lambda ^{*}(25x)]^{2}}}+2{\sqrt[{6}]{16\lambda ^{*}(x)^{2}\lambda ^{*}(25x)^{2}[1-\lambda ^{*}(x)^{2}][1-\lambda ^{*}(25x)^{2}]}}=1}](/media/api/rest_v1/media/math/render/svg/3756fcad97f5364f71c8cf7a78793d2b0eb9bd94)
![{\displaystyle {\sqrt[{4}]{\lambda ^{*}(x)\lambda ^{*}(49x)}}+{\sqrt[{8}]{[1-\lambda ^{*}(x)^{2}][1-\lambda ^{*}(49x)^{2}]}}=1}](/media/api/rest_v1/media/math/render/svg/aa90b0f70ea1967569b264855e7e44af5450b59e)
![{\displaystyle {\sqrt {\lambda ^{*}(x)\lambda ^{*}(121x)}}+{\sqrt[{4}]{[1-\lambda ^{*}(x)^{2}][1-\lambda ^{*}(121x)^{2}]}}+2{\sqrt[{12}]{16\lambda ^{*}(x)^{2}\lambda ^{*}(121x)^{2}[1-\lambda ^{*}(x)^{2}][1-\lambda ^{*}(121x)^{2}]}}=1}](/media/api/rest_v1/media/math/render/svg/933299309991833958ad7a1aaa4077c1d1e8719f)
![{\displaystyle {\sqrt[{4}]{\lambda ^{*}(x)\lambda ^{*}(529x)}}+{\sqrt[{8}]{[1-\lambda ^{*}(x)^{2}][1-\lambda ^{*}(529x)^{2}]}}+{\sqrt {2}}{\sqrt[{24}]{16\lambda ^{*}(x)^{2}\lambda ^{*}(529x)^{2}[1-\lambda ^{*}(x)^{2}][1-\lambda ^{*}(529x)^{2}]}}=1}](/media/api/rest_v1/media/math/render/svg/1ca31f5784b7c693a87c29c089e720a7eaa343fa)
![{\displaystyle G(x)=\sin\{2\arcsin[\lambda ^{*}(x)]\}^{-1/12}=1/[{\sqrt[{12}]{2\lambda ^{*}(x)}}{\sqrt[{24}]{1-\lambda ^{*}(x)^{2}}}]}](/media/api/rest_v1/media/math/render/svg/003c077da4a356e43ddb4095da8aa7969efa38f8)
![{\displaystyle g(x)=\tan\{2\arctan[\lambda ^{*}(x)]\}^{-1/12}={\sqrt[{12}]{[1-\lambda ^{*}(x)^{2}]/[2\lambda ^{*}(x)]}}}](/media/api/rest_v1/media/math/render/svg/0faafcdf649c59a5287a89cee5bf869a61f19229)
![{\displaystyle \lambda ^{*}(x)=\tan\{{\tfrac {1}{2}}\arctan[g(x)^{-12}]\}={\sqrt {g(x)^{24}+1}}-g(x)^{12}}](/media/api/rest_v1/media/math/render/svg/32925a11a5fe318f8e6304b291c922581578490e)
![{\displaystyle \lambda ^{*}(9)={\frac {1}{2}}({\sqrt {3}}-1)({\sqrt {2}}-{\sqrt[{4}]{3}})}](/media/api/rest_v1/media/math/render/svg/8ebf346c7efa5f6dae3efb593b460bd8a04a772f)
![{\displaystyle \lambda ^{*}(11)={\frac {1}{16}}({\sqrt {22}}+3{\sqrt {2}}){\bigl (}{\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{\bigr )}^{4}}](/media/api/rest_v1/media/math/render/svg/d69d49f896b81fec5f2d69225bdc5e09302b95b2)
![{\displaystyle \lambda ^{*}(16)=({\sqrt {2}}+1)^{2}({\sqrt[{4}]{2}}-1)^{4}}](/media/api/rest_v1/media/math/render/svg/10044e9406aa1541fe5dd161a1102b9e57bffe47)
![{\displaystyle \lambda ^{*}(17)={\frac {1}{4}}{\bigl (}{\sqrt {7+{\sqrt {17}}}}-{\sqrt {{\sqrt {17}}+3}}{\bigr )}{\bigl (}{\sqrt {5+{\sqrt {17}}}}-{\sqrt[{4}]{10{\sqrt {17}}+38}}{\bigr )}}](/media/api/rest_v1/media/math/render/svg/539d1c0bdefa21394f3ecfd3cb474e5a0d898576)
![{\displaystyle \lambda ^{*}(19)={\frac {1}{16}}(3{\sqrt {38}}+13{\sqrt {2}}){\bigl [}{\frac {1}{6}}({\sqrt {19}}-2+{\sqrt {3}}){\sqrt[{3}]{3{\sqrt {3}}-{\sqrt {19}}}}-{\frac {1}{6}}({\sqrt {19}}-2-{\sqrt {3}}){\sqrt[{3}]{3{\sqrt {3}}+{\sqrt {19}}}}-{\frac {1}{3}}(5-{\sqrt {19}}){\bigr ]}^{4}}](/media/api/rest_v1/media/math/render/svg/b9df7e0442001abb859b9f6175a2e9562a06ea52)
![{\displaystyle \lambda ^{*}(21)={\frac {1}{4}}{\bigl [}({\sqrt {7}}-{\sqrt {3}})({\sqrt {3}}+1){\sqrt {{\sqrt {7}}-2}}-2{\sqrt {14}}+5{\sqrt {2}}-{\sqrt {42}}+2{\sqrt {6}}{\bigr ]}}](/media/api/rest_v1/media/math/render/svg/38166ab18fe6598a3119b6833856af19f8e1e5b3)
![{\displaystyle \lambda ^{*}(23)={\frac {1}{32}}(5{\sqrt {2}}+{\sqrt {46}}){\bigl [}{\frac {2}{3}}+{\frac {1}{6}}({\sqrt {3}}+1){\sqrt[{3}]{100-12{\sqrt {69}}}}-{\frac {1}{6}}({\sqrt {3}}-1){\sqrt[{3}]{100+12{\sqrt {69}}}}{\bigr ]}^{4}}](/media/api/rest_v1/media/math/render/svg/3524267fc1d50e737443e9cfb8bb18ab56fce537)
![{\displaystyle \lambda ^{*}(24)=(2+{\sqrt {3}})^{2}{\bigl [}{\sqrt {3}}+{\sqrt {2}}-({\sqrt {3}}-1)({\sqrt {2}}+1){\sqrt {{\sqrt {3}}+{\sqrt {2}}}}{\bigr ]}^{2}}](/media/api/rest_v1/media/math/render/svg/b5d178c8b2f619d778dfb8a9b556b95e3e59c859)
![{\displaystyle \lambda ^{*}(25)={\frac {1}{2}}({\sqrt {10}}-2{\sqrt {2}})(3-2{\sqrt[{4}]{5}})}](/media/api/rest_v1/media/math/render/svg/8666dfbddc36725040a84cd147ad6b2ffedbcce6)
![{\displaystyle \lambda ^{*}(26)=({\sqrt {26}}+5)({\sqrt {2}}-1)^{2}\tan {\bigl [}{\tfrac {1}{4}}\pi -\arctan({\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}}){\bigr ]}^{4}}](/media/api/rest_v1/media/math/render/svg/0a27403a30e15545523e347df6e25ce99823b335)
![{\displaystyle \lambda ^{*}(30)=\tan {\bigl \{}{\tfrac {1}{2}}\arctan {\bigl [}({\sqrt {10}}-3)^{2}({\sqrt {5}}-2)^{2}{\bigr ]}{\bigr \}}}](/media/api/rest_v1/media/math/render/svg/850c757dfca3fa6262cdc2eb97386f449097621a)
![{\displaystyle \lambda ^{*}(34)=\tan {\bigl \{}{\tfrac {1}{4}}\arcsin {\bigl [}{\tfrac {1}{9}}({\sqrt {17}}-4)^{2}{\bigr ]}{\bigr \}}}](/media/api/rest_v1/media/math/render/svg/902a0ae0cdc4ade66988e9bf7864a47b786d8fa4)
![{\displaystyle \lambda ^{*}(42)=\tan {\bigl \{}{\tfrac {1}{2}}\arctan {\bigl [}(2{\sqrt {7}}-3{\sqrt {3}})^{2}(2{\sqrt {2}}-{\sqrt {7}})^{2}{\bigr ]}{\bigr \}}}](/media/api/rest_v1/media/math/render/svg/01791d22045830606fd6cb6a5c628542f3de2120)
![{\displaystyle \lambda ^{*}(46)=\tan {\bigl \{}{\tfrac {1}{4}}\arcsin {\bigl [}{\tfrac {1}{207}}(104{\sqrt {2}}-147){\bigr ]}{\bigr \}}}](/media/api/rest_v1/media/math/render/svg/cdbb1d758d12b91a7907bdabb33969e95a711f9c)
![{\displaystyle \lambda ^{*}(50)=({\sqrt {2}}-1)\tan {\bigl [}\arctan({\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}}}})-{\tfrac {1}{8}}\pi {\bigr ]}^{4}}](/media/api/rest_v1/media/math/render/svg/230c530ebf558a9025f378dff88fd5e4632dadce)
![{\displaystyle \lambda ^{*}(58)=\tan {\bigl [}{\tfrac {1}{4}}\arcsin({\tfrac {1}{9801}}){\bigr ]}}](/media/api/rest_v1/media/math/render/svg/9a6533f95de85a62c48ed37eb7ef9d39cb4c4a8b)
![{\displaystyle \lambda ^{*}({\frac {1}{4}})=2{\sqrt[{4}]{2}}({\sqrt {2}}-1)}](/media/api/rest_v1/media/math/render/svg/510a6a6052f24b66521c928f9d0fbfc319c61150)
![{\displaystyle \lambda ^{*}({\frac {3}{4}})={\sqrt[{4}]{8}}({\sqrt {3}}-{\sqrt {2}})({\sqrt {2}}+1){\sqrt {({\sqrt {3}}-1)^{3}}}}](/media/api/rest_v1/media/math/render/svg/3d0d814751f8549ee8f251d05ae750ec0ee2f971)
![{\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)]}}}](/media/api/rest_v1/media/math/render/svg/f8b90067a9ca00dcc538abd3ca0274689bb7631e)
![{\displaystyle {\frac {\mathrm {d} }{\mathrm {d} x}}{\frac {K({\sqrt {1-x^{2}}})^{2}}{K(x)^{2}}}=-{\frac {2K({\sqrt {1-x^{2}}})}{x(1-x^{2})K(x)^{3}}}{\bigl [}K(x)E({\sqrt {1-x^{2}}})+E(x)K({\sqrt {1-x^{2}}})-K(x)K({\sqrt {1-x^{2}}}){\bigr ]}}](/media/api/rest_v1/media/math/render/svg/bb7cee2680b29d11002cabd476fc738cc85ac0a6)
![{\displaystyle {\frac {\mathrm {d} }{\mathrm {d} x}}\lambda ^{*}(x)={\biggl [}{\frac {\mathrm {d} }{\mathrm {d} w}}\,{\frac {K({\sqrt {1-w^{2}}})^{2}}{K(w)^{2}}}{\biggr ]}^{-1}[w=\lambda ^{*}(x)]={\biggl [}-{\frac {\pi K({\sqrt {1-w^{2}}})}{w(1-w^{2})K(w)^{3}}}{\biggr ]}^{-1}[w=\lambda ^{*}(x)]=}](/media/api/rest_v1/media/math/render/svg/bfb6021b8b7c3daaca17e0072eb4e11664e41452)
![{\displaystyle ={\biggl [}-{\frac {w(1-w^{2})K(w)^{3}}{\pi K({\sqrt {1-w^{2}}})}}{\biggr ]}[w=\lambda ^{*}(x)]=-{\frac {\lambda ^{*}(x)\lambda ^{*}(1/x)^{2}K[\lambda ^{*}(x)]^{3}}{\pi K[\lambda ^{*}(1/x)]}}}](/media/api/rest_v1/media/math/render/svg/30d6193e276916e8d6c30c994f5e2a036d0b99dc)