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An Elementary Approach to Elliptic Integrals: Gaussian Averages and the Landen Transform

Ilhan M. Izmirli

Abstract


The Landen transform relies heavily on the concept of arithmetic-geometric mean. The arithmetic-geometric mean was discovered by Gauss when he was fourteen years old.

The purpose of this paper is to prove some basic properties of arithmetic-geometric mean of two numbers and then use these properties to find some numerical approximations to AGM(a0,b0) for certain specific values of a0 and b0 using nothing more than basic pre-calculus algebra.

We will than show that the most obvious manner in which the arithmetic-geometric mean can be applied to the computation of elliptic integrals is via the Landen transform and give an elementary proof of the Landen transform using nothing but basic trigonometry. Consequently, since we will be able to compute the values of elliptic integrals using nothing but this transform and the formula for AGM(a0,b0), and since both of these would be proved using elementary methods, we will establish an elementary method of computing elliptic integrals.

Keywords


Arithmetic-geometric mean, Landen Transform, the pendulum integral, elliptic integrals

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