Modular lambda function: Difference between revisions

Revision as of 21:13, 16 January 2022

Revision as of 05:06, 20 January 2022

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:<math>(1-u^8)(1-v^8)-(1-uv)^8=0.</math>

The quantities <math>u</math> and <math>v</math> have the following product representations which define them as [[Holomorphic function|holomorphic functions]] in the whole upper half-plane:

:<math>u=\sqrt{2}e^{p\pi i\tau/8}\prod_{k=1}^\infty \tanh\frac{(k-1+e^{2kp/2)\pi i\tau}}{1+e^{(2k-1)p\pi i\tau}},\quad v=\sqrt{2}e^{p\pi i\tau/8}\prod_{k=1}^\infty \frac{1+e^{2k2kp\pi i\tau}}{1+e^{(2k-1)p\pi i\tau}}.,</math>

:<math>v=\prod_{k=1}^\infty \tanh\frac{(k-1/2)\pi i}{\tau}=\sqrt{2}e^{\pi i\tau/8}\prod_{k=1}^\infty \frac{1+e^{2k\pi i\tau}}{1+e^{(2k-1)\pi i\tau}}.</math>

Since <math>\lambda(i)=1/2</math>, the modular equations can be used to give [[Algebraic number|algebraic values]] of <math>\lambda(pi)</math> for any prime <math>p</math>.<ref group="note">For any [[prime power]], we can iterate the modular equation of degree <math>p</math>. This process can be used to give algebraic values of <math>\lambda (ni)</math> for any <math>n\in\mathbb{N}.</math></ref> For any odd <math>n</math>, the algebraic values of <math>\lambda(ni)</math> are given by<ref name="Jacobi">{{Cite book |last1=Jacobi |first1=Carl Gustav Jacob |title=Fundamenta nova theoriae functionum ellipticarum|language=Latin|year=1829}} p. 42</ref><ref group="note"><math>\operatorname{sl}a\varpi</math> is algebraic for every <math>a\in\mathbb{Q}.</math></ref>

:<math>\lambda (ni)=\frac{1}{2^n}\prod_{k=1}^{n-1} \left(1-\operatorname{sl}^2\frac{k\varpi}{n}\right)^2</math>