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}[{\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)} 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(-\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 }} Weitere Identitäten zwischen Thetafunktion und Lambdafunktion [ Bearbeiten | Quelltext bearbeiten ] 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[-(a+{\tfrac {1}{2}})^{2}\pi {\sqrt {x}}]={\biggl \{}\sum _{a=-\infty }^{\infty }\operatorname {sech} [(a+{\tfrac {1}{2}})\pi {\sqrt {x}}]{\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} {\bigl [}(n-{\tfrac {1}{2}})\operatorname {arcosh} ({\tfrac {3}{2}}){\bigr ]}={\frac {\sqrt {5}}{4}}\sum _{a=-\infty }^{\infty }\operatorname {sech} {\bigl [}(a+{\tfrac {1}{2}})\operatorname {arcosh} ({\tfrac {3}{2}}){\bigr ]}=} = 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\{{\tfrac {1}{2}}\arcsin[\lambda ^{*}(x)]\}^{2}=\tanh\{{\tfrac {1}{2}}\operatorname {artanh} [\lambda ^{*}(x)]\}^{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)[1-\lambda ^{*}(x)^{2}][1-\lambda ^{*}(9x)^{2}]} 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}]{[1-\lambda ^{*}(x)^{2}][1-\lambda ^{*}(9x)^{2}]}}=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 [\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)]} 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}[1-\lambda ^{*}(x)^{2}][1-\lambda ^{*}(25x)^{2}]}}=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}]{[1-\lambda ^{*}(x)^{2}][1-\lambda ^{*}(49x)^{2}]}}=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}]{[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} 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}]{[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} 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\{{\tfrac {1}{2}}\arctan[g(x)^{-12}]\}={\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}}({\sqrt {6}}-{\sqrt {2}})} λ ∗ ( 4 ) = ( 2 − 1 ) 2 {\displaystyle \lambda ^{*}(4)=({\sqrt {2}}-1)^{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)=(2-{\sqrt {3}})({\sqrt {3}}-{\sqrt {2}})} λ ∗ ( 7 ) = 1 8 ( 3 2 − 14 ) {\displaystyle \lambda ^{*}(7)={\frac {1}{8}}(3{\sqrt {2}}-{\sqrt {14}})} 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}}({\sqrt {3}}-1)({\sqrt {2}}-{\sqrt[{4}]{3}})} λ ∗ ( 10 ) = ( 10 − 3 ) ( 2 − 1 ) 2 {\displaystyle \lambda ^{*}(10)=({\sqrt {10}}-3)({\sqrt {2}}-1)^{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}}({\sqrt {22}}+3{\sqrt {2}})\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)=({\sqrt {3}}-{\sqrt {2}})^{2}({\sqrt {2}}-1)^{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}}({\sqrt {10}}-{\sqrt {6}})(3-{\sqrt {5}})(2-{\sqrt {3}})} λ ∗ ( 16 ) = ( 2 + 1 ) 2 ( 2 4 − 1 ) 4 {\displaystyle \lambda ^{*}(16)=({\sqrt {2}}+1)^{2}({\sqrt[{4}]{2}}-1)^{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)=(2-{\sqrt {3}})^{2}({\sqrt {2}}-1)^{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}}(3{\sqrt {38}}+13{\sqrt {2}})\left[{\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}})\right]^{4}} λ ∗ ( 20 ) = ( 10 − 3 ) ( 5 + 2 ) ( 2 − 1 ) ( 5 − 1 − 1 ) 2 {\displaystyle \lambda ^{*}(20)=({\sqrt {10}}-3)({\sqrt {5}}+2)({\sqrt {2}}-1)\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[({\sqrt {7}}-{\sqrt {3}})({\sqrt {3}}+1){\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)=(10-3{\sqrt {11}})(3{\sqrt {11}}-7{\sqrt {2}})} 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}}(5{\sqrt {2}}+{\sqrt {46}})\left[{\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}}}}\right]^{4}} λ ∗ ( 24 ) = ( 2 + 3 ) 2 [ 3 + 2 − ( 3 − 1 ) ( 2 + 1 ) 3 + 2 ] 2 {\displaystyle \lambda ^{*}(24)=(2+{\sqrt {3}})^{2}\left[{\sqrt {3}}+{\sqrt {2}}-({\sqrt {3}}-1)({\sqrt {2}}+1){\sqrt {{\sqrt {3}}+{\sqrt {2}}}}\right]^{2}} λ ∗ ( 25 ) = 1 2 ( 10 − 2 2 ) ( 3 − 2 5 4 ) {\displaystyle \lambda ^{*}(25)={\frac {1}{2}}({\sqrt {10}}-2{\sqrt {2}})(3-2{\sqrt[{4}]{5}})} 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)=({\sqrt {26}}+5)({\sqrt {2}}-1)^{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 {\bigl \{}{\tfrac {1}{2}}\arctan {\bigl [}({\sqrt {10}}-3)^{2}({\sqrt {5}}-2)^{2}{\bigr ]}{\bigr \}}} λ ∗ ( 34 ) = tan { 1 4 arcsin [ 1 9 ( 17 − 4 ) 2 ] } {\displaystyle \lambda ^{*}(34)=\tan {\bigl \{}{\tfrac {1}{4}}\arcsin {\bigl [}{\tfrac {1}{9}}({\sqrt {17}}-4)^{2}{\bigr ]}{\bigr \}}} λ ∗ ( 42 ) = tan { 1 2 arctan [ ( 2 7 − 3 3 ) 2 ( 2 2 − 7 ) 2 ] } {\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 \}}} λ ∗ ( 46 ) = tan { 1 4 arcsin [ 1 207 ( 104 2 − 147 ) ] } {\displaystyle \lambda ^{*}(46)=\tan {\bigl \{}{\tfrac {1}{4}}\arcsin {\bigl [}{\tfrac {1}{207}}(104{\sqrt {2}}-147){\bigr ]}{\bigr \}}} 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)=({\sqrt {2}}-1)\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 {\bigl [}{\tfrac {1}{4}}\arcsin({\tfrac {1}{9801}}){\bigr ]}} In jener Liste sind die Lambda-Stern-Werte von Brüchen aufgelistet:
λ ∗ ( 1 2 ) = 2 2 − 2 {\displaystyle \lambda ^{*}({\frac {1}{2}})={\sqrt {2{\sqrt {2}}-2}}} λ ∗ ( 1 3 ) = 1 4 ( 6 + 2 ) {\displaystyle \lambda ^{*}({\frac {1}{3}})={\frac {1}{4}}({\sqrt {6}}+{\sqrt {2}})} λ ∗ ( 2 3 ) = ( 2 − 3 ) ( 3 + 2 ) {\displaystyle \lambda ^{*}({\frac {2}{3}})=(2-{\sqrt {3}})({\sqrt {3}}+{\sqrt {2}})} λ ∗ ( 1 4 ) = 2 2 4 ( 2 − 1 ) {\displaystyle \lambda ^{*}({\frac {1}{4}})=2{\sqrt[{4}]{2}}({\sqrt {2}}-1)} λ ∗ ( 3 4 ) = 8 4 ( 3 − 2 ) ( 2 + 1 ) ( 3 − 1 ) 3 {\displaystyle \lambda ^{*}({\frac {3}{4}})={\sqrt[{4}]{8}}({\sqrt {3}}-{\sqrt {2}})({\sqrt {2}}+1){\sqrt {({\sqrt {3}}-1)^{3}}}} λ ∗ ( 1 5 ) = 1 2 ( 5 − 1 + 3 − 5 ) {\displaystyle \lambda ^{*}({\frac {1}{5}})={\frac {1}{2}}\left({\sqrt {{\sqrt {5}}-1}}+{\sqrt {3-{\sqrt {5}}}}\right)} λ ∗ ( 2 5 ) = ( 10 − 3 ) ( 2 + 1 ) 2 {\displaystyle \lambda ^{*}({\frac {2}{5}})=({\sqrt {10}}-3)({\sqrt {2}}+1)^{2}} λ ∗ ( 3 5 ) = 1 16 ( 10 − 6 ) ( 3 + 5 ) ( 2 + 3 ) {\displaystyle \lambda ^{*}({\frac {3}{5}})={\frac {1}{16}}({\sqrt {10}}-{\sqrt {6}})(3+{\sqrt {5}})(2+{\sqrt {3}})} λ ∗ ( 4 5 ) = ( 10 + 3 ) ( 5 + 2 ) ( 2 + 1 ) ( 5 − 1 − 1 ) 2 {\displaystyle \lambda ^{*}({\frac {4}{5}})=({\sqrt {10}}+3)({\sqrt {5}}+2)({\sqrt {2}}+1)\left({\sqrt {{\sqrt {5}}-1}}-1\right)^{2}} 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)]}}} 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 . 
Van Wikipedia, de gratis encyclopedie
![{\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}[{\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}}}}](https://wikimedia.org/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}]}](https://wikimedia.org/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}]}](https://wikimedia.org/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}]}](https://wikimedia.org/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}}](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(-\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}](https://wikimedia.org/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}](https://wikimedia.org/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 }}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d0ad33fd26018f6f97eb11d98d31c918a8bc9a32)
![{\displaystyle \vartheta _{10}[\exp(-\pi {\sqrt {x}})]=\sum _{a=-\infty }^{\infty }\exp[-(a+{\tfrac {1}{2}})^{2}\pi {\sqrt {x}}]={\biggl \{}\sum _{a=-\infty }^{\infty }\operatorname {sech} [(a+{\tfrac {1}{2}})\pi {\sqrt {x}}]{\biggr \}}^{1/2}=}](https://wikimedia.org/api/rest_v1/media/math/render/svg/bf46e6a61bcd4423d72e1c07b3c58bf0d5dca421)
![{\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} {\bigl [}(n-{\tfrac {1}{2}})\operatorname {arcosh} ({\tfrac {3}{2}}){\bigr ]}={\frac {\sqrt {5}}{4}}\sum _{a=-\infty }^{\infty }\operatorname {sech} {\bigl [}(a+{\tfrac {1}{2}})\operatorname {arcosh} ({\tfrac {3}{2}}){\bigr ]}=}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f13c8212c6fac8ce71d8fd7c486bd5437d9ee22b)
![{\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\{{\tfrac {1}{2}}\arcsin[\lambda ^{*}(x)]\}^{2}=\tanh\{{\tfrac {1}{2}}\operatorname {artanh} [\lambda ^{*}(x)]\}^{2}}](https://wikimedia.org/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}]}](https://wikimedia.org/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}}](https://wikimedia.org/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}](https://wikimedia.org/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}}](https://wikimedia.org/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)]}](https://wikimedia.org/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}](https://wikimedia.org/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}](https://wikimedia.org/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}](https://wikimedia.org/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}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1ca31f5784b7c693a87c29c089e720a7eaa343fa)
![{\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\{{\tfrac {1}{2}}\arctan[g(x)^{-12}]\}={\sqrt {g(x)^{24}+1}}-g(x)^{12}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/32925a11a5fe318f8e6304b291c922581578490e)
![{\displaystyle \lambda ^{*}(9)={\frac {1}{2}}({\sqrt {3}}-1)({\sqrt {2}}-{\sqrt[{4}]{3}})}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8ebf346c7efa5f6dae3efb593b460bd8a04a772f)
![{\displaystyle \lambda ^{*}(11)={\frac {1}{16}}({\sqrt {22}}+3{\sqrt {2}})\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/95b7e06d62e8dd70ed56138476cc89c3bbd18eeb)
![{\displaystyle \lambda ^{*}(16)=({\sqrt {2}}+1)^{2}({\sqrt[{4}]{2}}-1)^{4}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/10044e9406aa1541fe5dd161a1102b9e57bffe47)
![{\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}}(3{\sqrt {38}}+13{\sqrt {2}})\left[{\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}})\right]^{4}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6e7984239c79707f204af8215e5b8c74fc1e6f9a)
![{\displaystyle \lambda ^{*}(21)={\frac {1}{4}}\left[({\sqrt {7}}-{\sqrt {3}})({\sqrt {3}}+1){\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/ad7a83a48f99850a63e52e9ac776c2374f49c9a9)
![{\displaystyle \lambda ^{*}(23)={\frac {1}{32}}(5{\sqrt {2}}+{\sqrt {46}})\left[{\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}}}}\right]^{4}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b25702328e499dc36c7e3a07bf5e83306d3f3b7d)
![{\displaystyle \lambda ^{*}(24)=(2+{\sqrt {3}})^{2}\left[{\sqrt {3}}+{\sqrt {2}}-({\sqrt {3}}-1)({\sqrt {2}}+1){\sqrt {{\sqrt {3}}+{\sqrt {2}}}}\right]^{2}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a390ab7a593226f58d8e3bb6b8fb717d5f17c0bd)
![{\displaystyle \lambda ^{*}(25)={\frac {1}{2}}({\sqrt {10}}-2{\sqrt {2}})(3-2{\sqrt[{4}]{5}})}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8666dfbddc36725040a84cd147ad6b2ffedbcce6)
![{\displaystyle \lambda ^{*}(26)=({\sqrt {26}}+5)({\sqrt {2}}-1)^{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/b3faa1b4710d22de7487baa7e4c5e68b9b390e7f)
![{\displaystyle \lambda ^{*}(30)=\tan {\bigl \{}{\tfrac {1}{2}}\arctan {\bigl [}({\sqrt {10}}-3)^{2}({\sqrt {5}}-2)^{2}{\bigr ]}{\bigr \}}}](https://wikimedia.org/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 \}}}](https://wikimedia.org/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 \}}}](https://wikimedia.org/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 \}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/cdbb1d758d12b91a7907bdabb33969e95a711f9c)
![{\displaystyle \lambda ^{*}(50)=({\sqrt {2}}-1)\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/24cf8cd704190b3f9bd39da556fe8cf4d6dd1427)
![{\displaystyle \lambda ^{*}(58)=\tan {\bigl [}{\tfrac {1}{4}}\arcsin({\tfrac {1}{9801}}){\bigr ]}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9a6533f95de85a62c48ed37eb7ef9d39cb4c4a8b)
![{\displaystyle \lambda ^{*}({\frac {1}{4}})=2{\sqrt[{4}]{2}}({\sqrt {2}}-1)}](https://wikimedia.org/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}}}}](https://wikimedia.org/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)]}}}](https://wikimedia.org/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 ]}}](https://wikimedia.org/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)]=}](https://wikimedia.org/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)]}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/30d6193e276916e8d6c30c994f5e2a036d0b99dc)