Content deleted Content added
Citation bot (talk | contribs) Alter: template type. Add: pages, s2cid, doi, issue, volume, date, journal. Removed proxy/dead URL that duplicated identifier. Removed parameters. Some additions/deletions were parameter name changes. | Use this bot. Report bugs. | #UCB_CommandLine |
mNo edit summary Tags: Visual edit Mobile edit Mobile web edit Advanced mobile edit |
||
| (3 intermediate revisions by 2 users not shown) | |||
Line 1:
{{short description|Symmetric holomorphic function}}
[[File:Modular lambda function in range -3 to 3.png|thumb|Modular lambda function in the complex plane.]]
In [[mathematics]], the '''modular lambda''' function λ(τ)<ref group="note><math>\lambda(\tau)</math> is not a [[Modular form#Modular functions|modular function]] (per the Wikipedia definition), but every modular function is a [[rational function]] in <math>\lambda(\tau)</math>. Some authors use a non-equivalent definition of "modular functions".</ref> is a highly symmetric [[
The q-expansion, where <math>q = e^{\pi i \tau}</math> is the [[Nome (mathematics)|nome]], is given by:
Line 102 ⟶ 103:
:<math>\lambda^*(x \in \mathbb{Q}^+) \in \mathbb{A}^+.</math>
<math>K(\lambda^*(x))</math> and <math>E(\lambda^*(x))</math> (the [[Elliptic integral#Complete elliptic integral of the second kind|complete elliptic integral of the second kind]]) can be expressed in closed form in terms of the [[gamma function]] for any <math>x\in\mathbb{Q}^+</math>, as Selberg and Chowla proved in 1949.<ref>{{Cite journal|title=On Epstein's Zeta Function (I).|last1=Chowla|first1=S.|last2=Selberg|first2=A.|journal=Proceedings of the National Academy of Sciences |date=1949 |volume=35 |issue=7 |page=373|doi=10.1073/PNAS.35.7.371 |s2cid=45071481 |doi-access=free|pmc=1063041}}</ref><ref>{{Cite web|url=https://eudml.org/doc/150803|title=On Epstein's Zeta-Function|last1=Chowla|first1=S.|last2=Selberg|first2=A.|website=EuDML|pages=86–110}}</ref>
The following expression is valid for all <math>n \in \mathbb{N}</math>:
| |||