π(PI) and the AGM: Gauss-Brent-Salamin Formula

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After a heavy dose of elliptic integral theory in the previous posts we can now prove the celebrated AGM formula for $\pi$ given independently by Gauss, Richard P. Brent and Eugene Salamin. So here we go

π(PI) and the AGM: Evaluating Elliptic Integrals contd.

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Complete Elliptic Integrals of Second Kind

After getting familiar with the AGM sequences and Landen Transformation, it is time to apply these concepts to evaluate elliptic integrals. Here we are going to focus on elliptic integrals of the second kind. To be more specific we are going to deal with $ J(a, b)$ defined by
$$ J(a, b) = \int_{0}^{\pi / 2}\sqrt{a^{2}\cos^{2}\theta + b^{2}\sin^{2}\theta}\,d\theta $$
Our strategy (well actually Landen's and Legendre's) here will be to analyze the defining integral under the Landen transformation
$$ \tan(\phi - \theta) = \frac{b}{a}\tan\theta $$

π(PI) and the AGM: Evaluating Elliptic Integrals

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Complete Elliptic Integrals of First Kind

In my earlier post I described the method for calculating complete elliptic integrals of first kind namely $ K(k)$ and $ I(a, b)$. To summarize the results we have
$$ K(k) = \int_{0}^{\pi / 2}\frac{d\theta}{\sqrt{1 - k^{2}\sin^{2}\theta}} = \frac{\pi}{2M(1, \sqrt{1 - k^{2}})} = \frac{\pi}{2M(1, k^{\prime})}$$ $$ I(a, b) = \int_{0}^{\pi / 2}\frac{d\theta}{\sqrt{a^{2}\cos^{2}\theta + b^{2}\sin^{2}\theta}} = \frac{\pi}{2M(a, b)}$$ where $ M(a, b)$ denotes the Arithmetic-Geometric Mean of two numbers $ a$ and $ b$ and $ k^{\prime}$ is the complementary modulus related to $ k$ by the following relation $$ k^{2} + k^{\prime 2} = 1$$

π(PI) and the AGM: Legendre's Identity

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While studying elliptic integrals (refer to previous post for an introduction to elliptic integrals) Legendre discovered a remarkable identity connecting the elliptic integrals of the first and second kinds. This identity at the same time connects these integrals to the mathematical constant $\pi$. This relation to $\pi$ was exploited by Gauss to derive a formula for $\pi$ based on AGM (which is the main topic of this series of posts).

π(PI) and the AGM: Introduction to Elliptic Integrals

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Introduction

In my last post I had described the adventures of Gauss with AGM (in this post AGM means Arithmetic-Geometric Mean, see the linked post). Gauss had established the deep connection between Elliptic Integrals and AGM and used his results in this field to ultimately derive a formula for calculating $\pi$ using AGM, but since it involved extraction of square roots it was not considered of much value in pre-computer era.

Arithmetic-Geometric Mean of Gauss

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Prelude

Contrary to the popular belief that mathematics is the most dreaded subject, many people in their younger years are struck by many mathematical curiosities. Some of them use these curiosities as puzzles for friends, others try to find the reason behind it and are satisfied once they find the reason. But the great heroes of mathematics are those who, being intrigued by a mathematical curiosity, develop the idea in a systematic manner and connect it to other ideas of existing mathematical knowledge.