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Landen's transformation is a mapping of the parameters of an elliptic integral, useful for the efficient numerical evaluation of elliptic functions. It was originally due to John Landen and independently rediscovered by Carl Friedrich Gauss.[1]
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But what is the Fourier Transform? A visual introduction.
Rotating polygons 180 degrees about their center | Transformations | Geometry | Khan Academy
*** Fouriertransformation bzw Fourierreihe Teil 1
11. Vorlesung Computergrafik SS 2014 - 3D Transformation - unistreams
Livestream zu DGL: Altklausur Differentialgleichungen für Ingenieure (Teil 2), TU Berlin
The transformation may be effected by integration by substitution. It is convenient to first cast the integral in an algebraic form by a substitution of , giving
A further substitution of gives the desired result
This latter step is facilitated by writing the radical as
and the infinitesimal as
so that the factor of is recognized and cancelled between the two factors.
Arithmetic-geometric mean and Legendre's first integral
If the transformation is iterated a number of times, then the parameters and converge very rapidly to a common value, even if they are initially of different orders of magnitude. The limiting value is called the arithmetic-geometric mean of and , . In the limit, the integrand becomes a constant, so that integration is trivial