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- Landen's_transformation abstract "Landen's transformation is a mapping of the parameters of an elliptic integral, which shows how the value of the integral, changes when its parameters:amplitude and modular angle changes following some dependency. As a special case, we can see when the transformation does not change the value of the integral.It was originally due to John Landen, although independently rediscovered by Carl Friedrich Gauss.For example,the incomplete elliptic integral of the first kind F is If parameters will be φ1 and k1 thenLanden's transformation shows how one can calculate second integral through first and formula connecting k and k1. A comprehensive list of transformations included in the tables 21.6-2 and 21.6-3 "Mathematical Handbook for scientists and engineers" by G.A. Korn and T.M. Korn. Accordingly 21.6-3 (aa)WhereConsider an example when the transformation does not change the value of the integral.Letand and are replaced by their arithmetic and geometric means respectively, that isObviouslyAccordingly formula (aa)(aaa) As follows from the formula (aaa)The same equation can be proved using a simple mathematical analysis.The transformation, may be achieved purely by integration by substitution. It is convenient to first cast the integral in an algebraic form by a substitution of , givingA further substitution of gives the desired result (in the algebraic form)This latter step is facilitated by writing the radical asand the infinitesimal as so that the factor of is easily recognized and cancelled between the two factors.".
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- Landen's_transformation wikiPageID "21424424".
- Landen's_transformation wikiPageRevisionID "586963917".
- Landen's_transformation hasPhotoCollection Landen's_transformation.
- Landen's_transformation subject Category:Elliptic_functions.
- Landen's_transformation type Abstraction100002137.
- Landen's_transformation type EllipticFunctions.
- Landen's_transformation type Function113783816.
- Landen's_transformation type MathematicalRelation113783581.
- Landen's_transformation type Relation100031921.
- Landen's_transformation comment "Landen's transformation is a mapping of the parameters of an elliptic integral, which shows how the value of the integral, changes when its parameters:amplitude and modular angle changes following some dependency.".
- Landen's_transformation label "Landen's transformation".
- Landen's_transformation label "Преобразование Ландена".
- Landen's_transformation sameAs m.05f9v9g.
- Landen's_transformation sameAs Q15615153.
- Landen's_transformation sameAs Q15615153.
- Landen's_transformation sameAs Landen's_transformation.
- Landen's_transformation wasDerivedFrom Landen's_transformation?oldid=586963917.
- Landen's_transformation isPrimaryTopicOf Landen's_transformation.