Definition als Lösung einer Integralgleichung

Die Elliptische Lambda-Funktion ausgedrückt mit einem Stern oben rechts über dem Lambda liefert den elliptischen Modul beziehungsweise die Exzentrizität auf folgende Weise:

${\displaystyle K\left[{\sqrt {1-\lambda ^{*}(x)^{2}}}\right]/K[\lambda ^{*}(x)]={\sqrt {x}}}$

Dabei bezeichnet K das vollständige elliptische Integral erster Art.

Die Funktionen Lambda und Lambda-Stern stehen in folgender Beziehung zueinander:

${\displaystyle \lambda ^{*}(x)={\sqrt {\lambda (i{\sqrt {x}})}}}$

Definitionen über die Jacobischen Thetafunktionen

Primär ist die Funktion λ*(x) so über die Theta-Nullwertfunktionen definiert:

${\displaystyle \lambda ^{*}(x)={\frac {\vartheta _{10}^{2}[\exp(-\pi {\sqrt {x}})]}{\vartheta _{00}^{2}[\exp(-\pi {\sqrt {x}})]}}}$

Ebenso kann Lambda-Stern-Funktion über den pythagoräisch komplementären Modul dargestellt werden:

${\displaystyle \lambda ^{*}(x)={\frac {\vartheta _{01}^{2}[\exp(-\pi /{\sqrt {x}})]}{\vartheta _{00}^{2}[\exp(-\pi /{\sqrt {x}})]}}}$

Auch über die Theta-Nicht-Nullwertfunktionen ist die Definition möglich:

${\displaystyle \lambda ^{*}(x)={\frac {\vartheta _{10}[{\tfrac {1}{4}}\pi$ ;\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:

${\displaystyle \vartheta _{00}(v;w)=\prod _{n=1}^{\infty }(1-w^{2n})[1+2\cos(2v)w^{2n-1}+w^{4n-2}]}$

${\displaystyle \vartheta _{01}(v;w)=\prod _{n=1}^{\infty }(1-w^{2n})[1-2\cos(2v)w^{2n-1}+w^{4n-2}]}$

${\displaystyle \vartheta _{10}(v;w)=2w^{1/4}\cos(v)\prod _{n=1}^{\infty }(1-w^{2n})[1+2\cos(2v)w^{2n}+w^{4n}]}$

Außerdem gelten folgende Ausdrucksweisen:

${\displaystyle \vartheta _{00}(w)=\vartheta _{00}(0;w)}$

${\displaystyle \vartheta _{01}(w)=\vartheta _{01}(0;w)}$

${\displaystyle \vartheta _{10}(w)=\vartheta _{10}(0;w)}$

Definitionen als Summenreihen und Produktreihen

Die Lambda-Stern-Werte können mit diesen sehr schnell konvergierenden Definitionsformeln berechnet werden:

${\displaystyle \lambda ^{*}(x)={\biggl \{}\sum _{a=-\infty }^{\infty }\exp {\bigl [}-(a+1/2)^{2}\pi {\sqrt {x}}{\bigr ]}{\biggr \}}^{2}{\biggl \{}\sum _{a=-\infty }^{\infty }\exp(-a^{2}\pi {\sqrt {x}}){\biggr \}}^{-2}}$

${\displaystyle \lambda ^{*}(x)={\biggl \{}\sum _{a=-\infty }^{\infty }\operatorname {sech} {\bigl [}(a+1/2)\pi {\sqrt {x}}{\bigr ]}{\biggr \}}{\biggl \{}\sum _{a=-\infty }^{\infty }\operatorname {sech} (a\pi {\sqrt {x}}){\biggr \}}^{-1}}$

${\displaystyle \lambda ^{*}(x)={\biggl [}\sum _{a=-\infty }^{\infty }(-1)^{a}\exp {\biggl (}-{\frac {a^{2}\pi }{\sqrt {x}}}{\biggr )}{\biggr ]}^{2}{\biggl [}\sum _{a=-\infty }^{\infty }\exp \left(-{\frac {a^{2}\pi }{\sqrt {x}}}\right){\biggr ]}^{-2}}$

${\displaystyle \lambda ^{*}(x)=\prod _{a=0}^{\infty }\operatorname {tanh} {\biggl [}{\frac {(a+1/2)\pi }{\sqrt {x}}}{\biggr ]}^{4}}$

Definition mit Integralen

Die Jacobische Theta-Nullwertfunktion ϑ₀₀ hat diese Integralidentität:

${\displaystyle \vartheta _{00}[\exp(-\pi {\sqrt {x}})]=1+2\exp(-\pi {\sqrt {x}})\int _{0}^{\infty }\exp(-\pi y^{2}){\frac {\exp(2\pi {\sqrt {x}})-\cos(2\pi {\sqrt[{4}]{x}}\,y)}{\cosh(2\pi {\sqrt {x}})-\cos(2\pi {\sqrt[{4}]{x}}\,y)}}\mathrm {d} y}$

${\displaystyle \vartheta _{00}[\exp(-2\pi {\sqrt {x}})]=1+2\exp(-2\pi {\sqrt {x}})\int _{0}^{\infty }\exp(-\pi y^{2}){\frac {\exp(4\pi {\sqrt {x}})-\cos(2\pi {\sqrt[{4}]{4x}}\,y)}{\cosh(4\pi {\sqrt {x}})-\cos(2\pi {\sqrt[{4}]{4x}}\,y)}}\mathrm {d} y}$

Die Lambda-Stern-Funktion kann dann auf jenem Definitionsweg dargestellt werden:

${\displaystyle \lambda ^{*}(x)=\sin {\biggl \langle }2\arccos {\biggl \{}{\frac {\vartheta _{00}[\exp(-2\pi {\sqrt {x}})]}{\vartheta _{00}[\exp(-\pi {\sqrt {x}})]}}{\biggr \}}{\biggr \rangle }}$

Weitere Identitäten zwischen Thetafunktion und Lambdafunktion

Für die Thetafunktionen ϑ₁₀ und ϑ₀₀ in reeller Form gelten folgende Formeln:

${\displaystyle \vartheta _{10}[\exp(-\pi {\sqrt {x}})]=\sum _{a=-\infty }^{\infty }\exp[-(a+{\tfrac {1}{2}})^{2}\pi {\sqrt {x}}]={\biggl \{}\sum _{a=-\infty }^{\infty }\operatorname {sech} [(a+{\tfrac {1}{2}})\pi {\sqrt {x}}]{\biggr \}}^{1/2}=}$

${\displaystyle ={\sqrt {2\pi ^{-1}\lambda ^{*}(x)K[\lambda ^{*}(x)]}}={\sqrt[{4}]{\lambda ^{*}(4x)}}{\sqrt {4\pi ^{-1}K[\lambda ^{*}(4x)]}}}$

${\displaystyle \vartheta _{00}[\exp(-\pi {\sqrt {x}})]=\sum _{a=-\infty }^{\infty }\exp(-a^{2}\pi {\sqrt {x}})={\biggl [}\sum _{a=-\infty }^{\infty }\operatorname {sech} (a\pi {\sqrt {x}}){\biggr ]}^{1/2}={\sqrt {2\pi ^{-1}K[\lambda ^{*}(x)]}}=\operatorname {agm} [1;\lambda ^{*}(1/x)]^{-1/2}}$

${\displaystyle \vartheta _{01}[\exp(-\pi {\sqrt {x}})]=\sum _{a=-\infty }^{\infty }(-1)^{a}\exp(-a^{2}\pi {\sqrt {x}})={\sqrt {2\pi ^{-1}\lambda ^{*}(1/x)K[\lambda ^{*}(x)]}}}$

Mit der Abkürzung agm wird das arithmetisch geometrische Mittel zum Ausdrück gebracht.

Von diesen beiden Thetafunktionen werden im Folgenden einige Theta-Nullwerte aufgelistet.

Anwendungsbeispiele

Unendliche Summe der Kehrwerte ungeradstelliger Fibonacci-Zahlen:

${\displaystyle \sum _{n=1}^{\infty }{\frac {1}{F_{2n-1}}}=\sum _{n=1}^{\infty }{\frac {\sqrt {5}}{2}}\operatorname {sech} {\bigl [}(n-{\tfrac {1}{2}})\operatorname {arcosh} ({\tfrac {3}{2}}){\bigr ]}={\frac {\sqrt {5}}{4}}\sum _{a=-\infty }^{\infty }\operatorname {sech} {\bigl [}(a+{\tfrac {1}{2}})\operatorname {arcosh} ({\tfrac {3}{2}}){\bigr ]}=}$

${\displaystyle ={\frac {\sqrt {5}}{4}}\vartheta _{10}(\Phi ^{-2})^{2}={\frac {\sqrt {5}}{8}}{\bigl [}\vartheta _{00}(\Phi ^{-1})^{2}-\vartheta _{01}(\Phi ^{-1})^{2}{\bigr ]}={\frac {\sqrt {5}}{\pi }}{\sqrt {\lambda ^{*}[16\,\pi ^{-2}\ln(\Phi )^{2}]}}K\{\lambda ^{*}[16\,\pi ^{-2}\ln(\Phi )^{2}]\}}$

Dabei ist ${\displaystyle \Phi ={\tfrac {{\sqrt {5}}+1}{2}}}$ die goldene Zahl.

Unendliche Summe der Kehrwerte ungeradstelliger Pell-Zahlen:

${\displaystyle \sum _{n=1}^{\infty }{\frac {1}{P(2n-1)}}={\frac {2{\sqrt {2}}}{\pi }}{\sqrt {\lambda ^{*}[16\,\pi ^{-2}\operatorname {arsinh} (1)^{2}]}}K\{\lambda ^{*}[16\,\pi ^{-2}\operatorname {arsinh} (1)^{2}]\}}$

Allgemeine Beziehungen

Generell ist jeder Lambda-Stern-Wert einer positiven rationalen Zahl eine positive algebraische Zahl:

${\displaystyle \lambda ^{*}(x\in \mathbb {Q} ^{+})\in \mathbb {A} ^{+}}$

Folgende Beziehung gilt für alle n ∈ 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:

${\displaystyle \lambda ^{*}(n^{2}x)=\lambda ^{*}(x)^{n}\prod _{a=1}^{n}\operatorname {sn} {\biggl \{}{\frac {2a-1}{n}}K{\bigl [}\lambda ^{*}(x){\bigr ]};\lambda ^{*}(x){\biggr \}}^{2}}$

${\displaystyle \lambda ^{*}(4n^{2})=\prod _{a=1}^{n}\operatorname {sl} \left({\frac {2a-1}{4n}}\varpi \right)^{4}}$

Hierbei ist sn die Jacobische elliptische Funktion Sinus Amplitudinis, während sl der lemniskatische Sinus ist.

Liste exemplarischer Beziehungen

Folgende weitere Beziehungen existieren zwischen den Lambda*-Funktionswerten:

${\displaystyle \lambda ^{*}(x)^{2}+\lambda ^{*}(1/x)^{2}=1}$

${\displaystyle [\lambda ^{*}(x)+1][\lambda ^{*}(4/x)+1]=2}$

${\displaystyle \lambda ^{*}(4x)={\frac {1-{\sqrt {1-\lambda ^{*}(x)^{2}}}}{1+{\sqrt {1-\lambda ^{*}(x)^{2}}}}}={\frac {\lambda ^{*}(x)^{2}}{{\bigl [}\,1+{\sqrt {1-\lambda ^{*}(x)^{2}}}\,{\bigr ]}^{2}}}}$

${\displaystyle \lambda ^{*}(4x)=\tan\{{\tfrac {1}{2}}\arcsin[\lambda ^{*}(x)]\}^{2}=\tanh\{{\tfrac {1}{2}}\operatorname {artanh} [\lambda ^{*}(x)]\}^{2}}$

${\displaystyle [\lambda ^{*}(x)-\lambda ^{*}(9x)]^{4}=16\lambda ^{*}(x)\lambda ^{*}(9x)[1-\lambda ^{*}(x)^{2}][1-\lambda ^{*}(9x)^{2}]}$

${\displaystyle \lambda ^{*}(x)-\lambda ^{*}(9x)=2[\lambda ^{*}(x)\lambda ^{*}(9x)]^{1/4}-2[\lambda ^{*}(x)\lambda ^{*}(9x)]^{3/4}}$

${\displaystyle {\sqrt {\lambda ^{*}(x)\lambda ^{*}(9x)}}+{\sqrt[{4}]{[1-\lambda ^{*}(x)^{2}][1-\lambda ^{*}(9x)^{2}]}}=1}$

${\displaystyle {\biggl [}{\frac {2\lambda ^{*}(x)}{1-\lambda ^{*}(x)^{2}}}{\biggr ]}^{1/2}-{\biggl [}{\frac {2\lambda ^{*}(25x)}{1-\lambda ^{*}(25x)^{2}}}{\biggr ]}^{1/2}=2{\biggl [}{\frac {2\lambda ^{*}(x)}{1-\lambda ^{*}(x)^{2}}}{\biggr ]}^{1/12}{\biggl [}{\frac {2\lambda ^{*}(25x)}{1-\lambda ^{*}(25x)^{2}}}{\biggr ]}^{1/12}+2{\biggl [}{\frac {2\lambda ^{*}(x)}{1-\lambda ^{*}(x)^{2}}}{\biggr ]}^{5/12}{\biggl [}{\frac {2\lambda ^{*}(25x)}{1-\lambda ^{*}(25x)^{2}}}{\biggr ]}^{5/12}}$

${\displaystyle [\lambda ^{*}(x)^{1/2}-\lambda ^{*}(25x)^{1/2}][\lambda ^{*}(x)+\lambda ^{*}(25x)+6\lambda ^{*}(x)^{1/2}\lambda ^{*}(25x)^{1/2}]=4\lambda ^{*}(x)^{1/4}\lambda ^{*}(25x)^{1/4}[1-\lambda ^{*}(x)\lambda ^{*}(25x)]}$

${\displaystyle \lambda ^{*}(x)\lambda ^{*}(25x)+{\sqrt {[1-\lambda ^{*}(x)]^{2}[1-\lambda ^{*}(25x)]^{2}}}+2{\sqrt[{6}]{16\lambda ^{*}(x)^{2}\lambda ^{*}(25x)^{2}[1-\lambda ^{*}(x)^{2}][1-\lambda ^{*}(25x)^{2}]}}=1}$

${\displaystyle {\sqrt[{4}]{\lambda ^{*}(x)\lambda ^{*}(49x)}}+{\sqrt[{8}]{[1-\lambda ^{*}(x)^{2}][1-\lambda ^{*}(49x)^{2}]}}=1}$

${\displaystyle {\sqrt {\lambda ^{*}(x)\lambda ^{*}(121x)}}+{\sqrt[{4}]{[1-\lambda ^{*}(x)^{2}][1-\lambda ^{*}(121x)^{2}]}}+2{\sqrt[{12}]{16\lambda ^{*}(x)^{2}\lambda ^{*}(121x)^{2}[1-\lambda ^{*}(x)^{2}][1-\lambda ^{*}(121x)^{2}]}}=1}$

${\displaystyle {\sqrt[{4}]{\lambda ^{*}(x)\lambda ^{*}(529x)}}+{\sqrt[{8}]{[1-\lambda ^{*}(x)^{2}][1-\lambda ^{*}(529x)^{2}]}}+{\sqrt {2}}{\sqrt[{24}]{16\lambda ^{*}(x)^{2}\lambda ^{*}(529x)^{2}[1-\lambda ^{*}(x)^{2}][1-\lambda ^{*}(529x)^{2}]}}=1}$

Ramanujansche Funktionen

Folgende Beziehungen gelten zu den Ramanujanschen Funktionen g und G:

${\displaystyle G(x)=\sin\{2\arcsin[\lambda ^{*}(x)]\}^{-1/12}=1\left/\left[{\sqrt[{12}]{2\lambda ^{*}(x)}}{\sqrt[{24}]{1-\lambda ^{*}(x)^{2}}}\right]\right.}$

${\displaystyle g(x)=\tan\{2\arctan[\lambda ^{*}(x)]\}^{-1/12}={\sqrt[{12}]{[1-\lambda ^{*}(x)^{2}]/[2\lambda ^{*}(x)]}}}$

${\displaystyle \lambda ^{*}(x)=\tan\{{\tfrac {1}{2}}\arctan[g(x)^{-12}]\}={\sqrt {g(x)^{24}+1}}-g(x)^{12}}$

Lambda-Stern-Werte ganzer Zahlen

In dieser Liste werden die Lambda-Stern-Werte der ganzen Zahlen 1 bis 25 radikalisch dargestellt:

${\displaystyle \lambda ^{*}(1)={\frac {1}{2}}{\sqrt {2}}}$

${\displaystyle \lambda ^{*}(2)={\sqrt {2}}-1}$

${\displaystyle \lambda ^{*}(3)={\frac {1}{4}}({\sqrt {6}}-{\sqrt {2}})}$

${\displaystyle \lambda ^{*}(4)=({\sqrt {2}}-1)^{2}}$

${\displaystyle \lambda ^{*}(5)={\frac {1}{2}}\left({\sqrt {{\sqrt {5}}-1}}-{\sqrt {3-{\sqrt {5}}}}\right)}$

${\displaystyle \lambda ^{*}(6)=(2-{\sqrt {3}})({\sqrt {3}}-{\sqrt {2}})}$

${\displaystyle \lambda ^{*}(7)={\frac {1}{8}}(3{\sqrt {2}}-{\sqrt {14}})}$

${\displaystyle \lambda ^{*}(8)=\left({\sqrt {2}}+1-{\sqrt {2{\sqrt {2}}+2}}\right)^{2}}$

${\displaystyle \lambda ^{*}(9)={\frac {1}{2}}({\sqrt {3}}-1)({\sqrt {2}}-{\sqrt[{4}]{3}})}$

${\displaystyle \lambda ^{*}(10)=({\sqrt {10}}-3)({\sqrt {2}}-1)^{2}}$

${\displaystyle \lambda ^{*}(11)={\frac {1}{16}}({\sqrt {22}}+3{\sqrt {2}})\left({\frac {1}{3}}{\sqrt[{3}]{6{\sqrt {3}}+2{\sqrt {11}}}}-{\frac {1}{3}}{\sqrt[{3}]{6{\sqrt {3}}-2{\sqrt {11}}}}+{\frac {1}{3}}{\sqrt {11}}-1\right)^{4}}$

${\displaystyle \lambda ^{*}(12)=({\sqrt {3}}-{\sqrt {2}})^{2}({\sqrt {2}}-1)^{2}}$

${\displaystyle \lambda ^{*}(13)={\frac {1}{2}}\left({\sqrt {5{\sqrt {13}}-17}}-{\sqrt {19-5{\sqrt {13}}}}\right)}$

${\displaystyle \lambda ^{*}(14)=\left(2{\sqrt {2}}+2-{\sqrt {8{\sqrt {2}}+11}}\right)\left({\sqrt {8{\sqrt {2}}+11}}-{\sqrt {8{\sqrt {2}}+10}}\right)}$

${\displaystyle \lambda ^{*}(15)={\frac {1}{16}}({\sqrt {10}}-{\sqrt {6}})(3-{\sqrt {5}})(2-{\sqrt {3}})}$

${\displaystyle \lambda ^{*}(16)=({\sqrt {2}}+1)^{2}({\sqrt[{4}]{2}}-1)^{4}}$

${\displaystyle \lambda ^{*}(17)={\frac {1}{4}}\left({\sqrt {7+{\sqrt {17}}}}-{\sqrt {{\sqrt {17}}+3}}\right)\left({\sqrt {5+{\sqrt {17}}}}-{\sqrt[{4}]{10{\sqrt {17}}+38}}\right)}$

${\displaystyle \lambda ^{*}(18)=(2-{\sqrt {3}})^{2}({\sqrt {2}}-1)^{3}}$

${\displaystyle \lambda ^{*}(19)={\frac {1}{16}}(3{\sqrt {38}}+13{\sqrt {2}})\left[{\frac {1}{6}}({\sqrt {19}}-2+{\sqrt {3}}){\sqrt[{3}]{3{\sqrt {3}}-{\sqrt {19}}}}-{\frac {1}{6}}({\sqrt {19}}-2-{\sqrt {3}}){\sqrt[{3}]{3{\sqrt {3}}+{\sqrt {19}}}}-{\frac {1}{3}}(5-{\sqrt {19}})\right]^{4}}$

${\displaystyle \lambda ^{*}(20)=({\sqrt {10}}-3)({\sqrt {5}}+2)({\sqrt {2}}-1)\left({\sqrt {{\sqrt {5}}-1}}-1\right)^{2}}$

${\displaystyle \lambda ^{*}(21)={\frac {1}{4}}\left[({\sqrt {7}}-{\sqrt {3}})({\sqrt {3}}+1){\sqrt {{\sqrt {7}}-2}}-2{\sqrt {14}}+5{\sqrt {2}}-{\sqrt {42}}+2{\sqrt {6}}\right]}$

${\displaystyle \lambda ^{*}(22)=(10-3{\sqrt {11}})(3{\sqrt {11}}-7{\sqrt {2}})}$

${\displaystyle \lambda ^{*}(23)={\frac {1}{32}}(5{\sqrt {2}}+{\sqrt {46}})\left[{\frac {2}{3}}+{\frac {1}{6}}({\sqrt {3}}+1){\sqrt[{3}]{100-12{\sqrt {69}}}}-{\frac {1}{6}}({\sqrt {3}}-1){\sqrt[{3}]{100+12{\sqrt {69}}}}\right]^{4}}$

${\displaystyle \lambda ^{*}(24)=(2+{\sqrt {3}})^{2}\left[{\sqrt {3}}+{\sqrt {2}}-({\sqrt {3}}-1)({\sqrt {2}}+1){\sqrt {{\sqrt {3}}+{\sqrt {2}}}}\right]^{2}}$

${\displaystyle \lambda ^{*}(25)={\frac {1}{2}}({\sqrt {10}}-2{\sqrt {2}})(3-2{\sqrt[{4}]{5}})}$

Weitere Lambdafunktionswerte des Schemas λ*(4n - 2) mit n ∈ ℕ können vereinfacht mit dem Tangens dargestellt werden:

${\displaystyle \lambda ^{*}(26)=({\sqrt {26}}+5)({\sqrt {2}}-1)^{2}\tan \left[{\tfrac {1}{4}}\pi -\arctan \left({\tfrac {1}{3}}{\sqrt[{3}]{3{\sqrt {3}}+{\sqrt {26}}}}-{\tfrac {1}{3}}{\sqrt[{3}]{3{\sqrt {3}}-{\sqrt {26}}}}+{\tfrac {1}{6}}{\sqrt {26}}-{\tfrac {1}{2}}{\sqrt {2}}\right)\right]^{4}}$

${\displaystyle \lambda ^{*}(30)=\tan {\bigl \{}{\tfrac {1}{2}}\arctan {\bigl [}({\sqrt {10}}-3)^{2}({\sqrt {5}}-2)^{2}{\bigr ]}{\bigr \}}}$

${\displaystyle \lambda ^{*}(34)=\tan {\bigl \{}{\tfrac {1}{4}}\arcsin {\bigl [}{\tfrac {1}{9}}({\sqrt {17}}-4)^{2}{\bigr ]}{\bigr \}}}$

${\displaystyle \lambda ^{*}(42)=\tan {\bigl \{}{\tfrac {1}{2}}\arctan {\bigl [}(2{\sqrt {7}}-3{\sqrt {3}})^{2}(2{\sqrt {2}}-{\sqrt {7}})^{2}{\bigr ]}{\bigr \}}}$

${\displaystyle \lambda ^{*}(46)=\tan {\bigl \{}{\tfrac {1}{4}}\arcsin {\bigl [}{\tfrac {1}{207}}(104{\sqrt {2}}-147){\bigr ]}{\bigr \}}}$

${\displaystyle \lambda ^{*}(50)=({\sqrt {2}}-1)\tan \left[\arctan \left({\tfrac {1}{3}}{\sqrt {5}}-{\tfrac {1}{3}}{\sqrt[{3}]{6{\sqrt {30}}+4{\sqrt {5}}}}+{\tfrac {1}{3}}{\sqrt[{3}]{6{\sqrt {30}}-4{\sqrt {5}}}}\right)-{\tfrac {1}{8}}\pi \right]^{4}}$

${\displaystyle \lambda ^{*}(58)=\tan {\bigl [}{\tfrac {1}{4}}\arcsin({\tfrac {1}{9801}}){\bigr ]}}$

Lambda-Stern-Werte von gebrochen rationalen Zahlen

In jener Liste sind die Lambda-Stern-Werte von Brüchen aufgelistet:

${\displaystyle \lambda ^{*}({\frac {1}{2}})={\sqrt {2{\sqrt {2}}-2}}}$

${\displaystyle \lambda ^{*}({\frac {1}{3}})={\frac {1}{4}}({\sqrt {6}}+{\sqrt {2}})}$

${\displaystyle \lambda ^{*}({\frac {2}{3}})=(2-{\sqrt {3}})({\sqrt {3}}+{\sqrt {2}})}$

${\displaystyle \lambda ^{*}({\frac {1}{4}})=2{\sqrt[{4}]{2}}({\sqrt {2}}-1)}$

${\displaystyle \lambda ^{*}({\frac {3}{4}})={\sqrt[{4}]{8}}({\sqrt {3}}-{\sqrt {2}})({\sqrt {2}}+1){\sqrt {({\sqrt {3}}-1)^{3}}}}$

${\displaystyle \lambda ^{*}({\frac {1}{5}})={\frac {1}{2}}\left({\sqrt {{\sqrt {5}}-1}}+{\sqrt {3-{\sqrt {5}}}}\right)}$

${\displaystyle \lambda ^{*}({\frac {2}{5}})=({\sqrt {10}}-3)({\sqrt {2}}+1)^{2}}$

${\displaystyle \lambda ^{*}({\frac {3}{5}})={\frac {1}{16}}({\sqrt {10}}-{\sqrt {6}})(3+{\sqrt {5}})(2+{\sqrt {3}})}$

${\displaystyle \lambda ^{*}({\frac {4}{5}})=({\sqrt {10}}+3)({\sqrt {5}}+2)({\sqrt {2}}+1)\left({\sqrt {{\sqrt {5}}-1}}-1\right)^{2}}$