Modular lambda function: Difference between revisions

Revision as of 00:49, 4 January 2022

Revision as of 16:53, 6 January 2022

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

==Lambda-star==

==Elliptic modulus==

[[File:Lambda star function in range -3 to 3.png|thumb|λ*(x) in the complex plane.]]

===Definition and computation of lambda-star===

The function λ*(x)<ref>{{Cite book |last1=Borwein |first1=Jonathan M. |last2=Borwein| first2=Peter B. |title=Pi and the AGM: A Study in Analytic Number Theory and Computational Complexity |publisher=Wiley-Interscience |year=1987 |edition=First |isbn=0-471-83138-7}} p. 152</ref> gives the value of the elliptic modulus <math>k</math>, for which the complete [[elliptic integral]] of the first kind <math>K(k)</math> and its complementary counterpart <math>K\left(\sqrt{1-k^2}\right)</math> are related by following expression:

:<math>\frac{K\left[\sqrt{1-\lambda^*(x)^2}\right]}{K[\lambda^*(x)]} = \sqrt{x}</math>