Elementary Approach to Modular Equations: Ramanujan's Theory 3

Connection between Theta Functions and Hypergeometric Functions

Let's recall the Gauss Transformation formula from an earlier post: $$F\left(a, b; 2b; \frac{4x}{(1 + x)^{2}}\right) = (1 + x)^{2a}F\left(a, a - b + \frac{1}{2}; b + \frac{1}{2}; x^{2}\right)$$ where $ F$ is the hypergeometric function $ {}_{2}F_{1}$. Putting $ a = b = 1/2$ we get $${}_{2}F_{1}\left(\frac{1}{2}, \frac{1}{2}; 1; \frac{4x}{(1 + x)^{2}}\right) = (1 + x)\,{}_{2}F_{1}\left(\frac{1}{2}, \frac{1}{2}; 1; x^{2}\right)$$ or $${}_{2}F_{1}\left(\frac{1}{2}, \frac{1}{2}; 1; 1 - \left(\frac{1 - x}{1 + x}\right)^{2}\right) = (1 + x)\,{}_{2}F_{1}\left(\frac{1}{2}, \frac{1}{2}; 1; x^{2}\right)$$

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Elementary Approach to Modular Equations: Ramanujan's Theory 2

Ramanujan's Theory of Elliptic Functions

Ramanujan used the letter $ x$ in place of $ k^{2}$ and studied the function $ {}_{2}F_{1}(1/2, 1/2; 1; x)$ in great detail and developed his theory of elliptic integrals and functions.

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Elementary Approach to Modular Equations: Ramanujan's Theory 1

Ramanujan developed his theory of modular equations using the theory of theta functions independently of Jacobi. A complete understanding of his approach is unfortunately not possible till now because he did not publish something like Fundamenta Nova containing detailed explanations of his approach. What we have today is his Notebooks edited by Bruce C. Berndt and his Collected Papers. His Notebooks are just statements of various mathematical formulas without any proof. A large part of these notebooks is concerned with modular equations and modern authors have not been able to discern his methods fully. Hence I will not be able to present a true picture of his approach. Rather I will try to present whatever I understand from his Collected Papers and his Notebooks and only focus on the elementary aspects.

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Elementary Approach to Modular Equations: Jacobi's Transformation Theory 5

Jacobi's Second Real Transformation

The second transformation is obtained by taking $ m = 0, m' = 1$ so that $ \omega = iK'/p$. On the face of it the transformation thus involves imaginary quantities, but will be shown later to be a real transformation only. In the case of this transformation we will use $ l_{1}$ in place of $ l$ and $ M_{1}$ in place of $ M$. Also we will keep the factor $ (-1)^{(p - 1)/2}$ with the multiplier $ M_{1}$. We thus obtain the following by putting $ \omega = iK'/p$ in the general formulas in $ 2s\omega$: \begin{align}\text{sn}\left(\frac{u}{M_{1}}, l_{1}\right) &= \frac{\text{sn}\,u}{M_{1}}\prod_{s = 1}^{(p - 1)/2}\dfrac{1 - \dfrac{\text{sn}^{2}\,u}{\text{sn}^{2}\,\dfrac{2siK'}{p}}}{1 - k^{2}\,\text{sn}^{2}\,u\,\text{sn}^{2}\,\dfrac{2siK'}{p}}\notag\\ &= \sqrt{\frac{k^{p}}{l_{1}}}\prod_{s = -(p - 1)/2}^{(p - 1)/2}\text{sn}\left(u + \frac{2siK'}{p}\right)\notag\\ \text{cn}\left(\frac{u}{M_{1}}, l_{1}\right) &= \text{cn}\,u\prod_{s = 1}^{(p - 1)/2}\dfrac{1 - \dfrac{\text{sn}^{2}\,u}{\text{sn}^{2}\left(K - \dfrac{2siK'}{p}\right)}}{1 - k^{2}\,\text{sn}^{2}\,u\,\text{sn}^{2}\,\dfrac{2siK'}{p}}\notag\\ &= \sqrt{\frac{l'_{1}k^{p}}{l_{1}k'^{p}}}\prod_{s = -(p - 1)/2}^{(p - 1)/2}\text{cn}\left(u + \frac{2siK'}{p}\right)\notag\\ \text{dn}\left(\frac{u}{M_{1}}, l_{1}\right) &= \text{dn}\,u\prod_{s = 1}^{(p - 1)/2}\dfrac{1 - k^{2}\,\text{sn}^{2}\,u\,\text{sn}^{2}\left(K - \dfrac{2siK'}{p}\right)}{1 - k^{2}\,\text{sn}^{2}\,u\,\text{sn}^{2}\,\dfrac{2siK'}{p}}\notag\\ &= \sqrt{\frac{l'_{1}}{k'^{p}}}\prod_{s = -(p - 1)/2}^{(p - 1)/2}\text{dn}\left(u + \frac{2siK'}{p}\right)\notag\\ M_{1} &= (-1)^{(p - 1)/2}\prod_{s = 1}^{(p - 1)/2}\left(\dfrac{\text{sn}\left(K - \dfrac{2siK'}{p}\right)}{\text{sn}\,\dfrac{2siK'}{p}}\right)^{2}\notag\\ l_{1} &= k^{p}\prod_{s = 1}^{(p - 1)/2}\text{sn}^{4}\left(K - \frac{2siK'}{p}\right)\notag\\ l'_{1} &= \dfrac{k'^{p}}{{\displaystyle\prod_{s = 1}^{(p - 1)/2}\text{dn}^{4}\,\frac{2siK'}{p}}}\notag\end{align}

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Elementary Approach to Modular Equations: Jacobi's Transformation Theory 4

Transformation of Elliptic Functions

The relation $$y = \frac{x}{M}\prod_{s = 1}^{(p - 1)/2}\dfrac{1 - \dfrac{x^{2}}{\text{sn}^{2}\,4s\omega}}{1 - k^{2}x^{2}\text{sn}^{2}\,4s\omega}$$ and other variants of it \begin{align}1 - y &= (1 - x)\prod_{s = 1}^{(p - 1)/2}\dfrac{\left(1 - \dfrac{x}{\text{sn}(K - 4s\omega)}\right)^{2}}{1 - k^{2}x^{2}\,\text{sn}^{2}\,4s\omega}\notag\\ 1 + y &= (1 + x)\prod_{s = 1}^{(p - 1)/2}\dfrac{\left(1 + \dfrac{x}{\text{sn}(K - 4s\omega)}\right)^{2}}{1 - k^{2}x^{2}\,\text{sn}^{2}\,4s\omega}\notag\\ 1 - ly &= (1 - kx)\prod_{s = 1}^{(p - 1)/2}\frac{(1 - kx\,\text{sn}(K - 4s\omega))^{2}}{1 - k^{2}x^{2}\,\text{sn}^{2}\,4s\omega}\notag\\ 1 + ly &= (1 + kx)\prod_{s = 1}^{(p - 1)/2}\frac{(1 + kx\,\text{sn}(K - 4s\omega))^{2}}{1 - k^{2}x^{2}\,\text{sn}^{2}\,4s\omega}\notag\end{align} as described in previous post lead to the differential equation $$\frac{Mdy}{\sqrt{(1 - y^{2})(1 - l^{2}y^{2})}} = \frac{dx}{\sqrt{(1 - x^{2})(1 - k^{2}x^{2})}}$$ when $$M = (-1)^{(p - 1)/2}\prod_{s = 1}^{(p - 1)/2}\left(\frac{\text{sn}(K - 4s\omega)}{\text{sn}\,4s\omega}\right)^{2}$$ and $$l = k^{p}\prod_{s = 1}^{(p - 1)/2}\text{sn}^{4}(K - 4s\omega)$$

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Elementary Approach to Modular Equations: Jacobi's Transformation Theory 3

Analytic Approach to Transformation Theory

Jacobi understood that the algebraic approach for obtaining modular equations could not be applied easily in case of higher degrees. Hence he followed an analytic approach. The idea he used was to express the relation $ y = xN(1, x^{2})/D(1, x^{2})$ in a form where each of $ N$ and $ D$ appears as a product of various factors. Essentially he examined the roots of $ N, D$ and expressed them in form of a product where each factor corresponds to a given root. This approach was very useful for Jacobi as he used this relation to finally develop the theory of Theta Functions and their relation to elliptic functions. In fact the entire Fundamenta Nova is split into two sections, the first section dealing with transformation theory and the second section dealing with the expansion of elliptic functions into infinite series and products (which is basically the theory of theta functions).

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Elementary Approach to Modular Equations: Jacobi's Transformation Theory 2

In this post we will apply the technique described in previous post to obtain modular equations of degree $ 3$ and $ 5$.

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Elementary Approach to Modular Equations: Jacobi's Transformation Theory 1

To recapitulate the basics of elliptic integral theory (details here) we have $$K = K(k) = \int_{0}^{\pi/2}\frac{d\theta}{\sqrt{1 - k^{2}\sin^{2}\theta}} = \int_{0}^{1}\frac{dx}{\sqrt{(1 - x^{2})(1 - k^{2}x^{2})}}$$ $$E = E(k) = \int_{0}^{\pi/2}\sqrt{1 - k^{2}\sin^{2}\theta}\,d\theta = \int_{0}^{1}\frac{\sqrt{1 - k^{2}x^{2}}}{\sqrt{1 - x^{2}}}\,dx$$

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Elementary Approach to Modular Equations: Hypergeometric Series 2

To continue our adventures (which started here) with the hypergeometric function we are going to establish the following identity
If $ a + b + (1/2)$ is neither zero nor a negative integer and if $ |x| < 1$ and $ |4x(1 - x)| < 1$, then $$F\left(a, b; a + b + \frac{1}{2}; 4x(1 - x)\right) = F\left(2a, 2b; a + b + \frac{1}{2}; x\right)$$

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Elementary Approach to Modular Equations: Hypergeometric Series 1

For quite some time I have been studying Ramanujan's Modular Equations and Approximations to $ \pi$ and in this series of posts I will try to present my understanding of the modular equations. Ramanujan's work on modular equations was brought to limelight by Borwein brothers in their famous book Pi and the AGM and later on by Bruce C. Berndt through Ramanujan Notebooks. Much of what I present here would also be based on the material presented in these books. However my approach here is going to be elementary and requires at best a working knowledge of calculus. Apart from this reader is expected to have some background on elliptic functions and theta functions as presented in my previous series of posts (here and here).

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Elliptic Functions: Landen's Transformation

In our introductory post we had talked about the similarities of the elliptic functions with the circular functions. At the same time we discussed the properties which were quite unlike those of circular functions (like double periodicity). In this post we are going to discuss some further properties of elliptic functions which have no analogue in the theory of circular functions.

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Elliptic Functions: Complex Variables

Introduction

So far we have studied the elliptic functions of real variables, i.e. we consider the $ u$ in $ \text{sn}(u, k)$ to be a real number and have so far found that they have properties similar to the circular functions (for example they are bounded and periodic). However their real nature and power is exhibited only when we go in the realm of complex numbers and study them as functions of complex variables.

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Elliptic Functions: Introduction

Introduction

In the previous posts we have covered introductory material on the following topics like elliptic integrals, AGM, and theta functions. All the concepts are tightly coupled with each other and belong more properly to the theory of elliptic functions. The theory of elliptic functions puts all the above concepts into a unified perspective and provides us a coherent picture. The approach to elliptic functions would be again very introductory and we will not pursue the topics related to "theory of functions of complex variable" in detail.

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The Magic of Theta Functions: Contd.

In the previous post we studied some interesting properties of theta functions which were used to relate them to AGM and thereby to elliptic integrals. We will continue to explore further in this direction and start with a remarkable property of theta function $ \theta_{3}(q)$.

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The Magic of Theta Functions

Introduction

Theta functions were originally introduced by Carl Gustav Jacob Jacobi while studying elliptic functions (which are in turn related to elliptic integrals). These functions are also connected with number theory and they have many interesting properties besides. Since they are related to elliptic integrals and we have seen in a previous post that the elliptic integrals are related to the AGM (arithmetic-geometric mean), it follows that the theta functions are related to the AGM. We will cover these topics in this series of posts and will also mention some number theoretic applications of theta functions.

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

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

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

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

$$ \tan(\phi - \theta) = \frac{b}{a}\tan\theta $$

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π(PI) and the AGM: Evaluating Elliptic Integrals

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 $$ k^{2} + k^{\prime 2} = 1$$

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

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).

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

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.

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