Ramanujan's take on Chudnovsky series for 1/π(PI): Part 2

In the previous post we have handled the evaluation of $P_n=P(-e^{-\pi\sqrt{n}}) $ for $n=11,27$. We will evaluate $P_n$ for $n=19,35$ in the current post and also discuss an empirical approach for $n=43,67,163$. Finally we will use the information in table given by Ramanujan to obtain certain series for $1/\pi$ (including the famous one by Chudnovsky brothers).

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Ramanujan's take on Chudnovsky series for 1/π(PI): Part 1

We have discussed a proof of Chudnovsky series for $1/\pi$ in this post based on Ramanujan's ideas as presented in this post. However a serious look at one of the pages from his lost notebook suggests that Ramanujan used a slightly different approach to obtain Chudnovsky type series and he also performed all the desired calculations needed to get the series in explicit form. This is what we intend to discuss in the current post.

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Modular Equations and Approximations to π(PI): Part 3

Series Based on Alternative Theories

In the previous post we established certain series for $1/\pi$ following Ramanujan's technique. These were based on formulas in the classical theory of elliptic functions and integrals. In the field of elliptic functions, Ramanujan surpassed all his predecessors and developed alternative theories which bore striking resemblance to the classical theory and thus provided a grand generalization of the theory of elliptic functions.

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Modular Equations and Approximations to π(PI): Part 2

Ramanujan's Series for $ \pi$

Using the values of the function $ P(q)$ for $ q = e^{-\pi\sqrt{n}}$ (see previous post for the definition of $ P(q)$) Ramanujan was able to derive many beautiful series for $ \pi$. He did this in very clever way. The fundamental idea he used was the fact that the function $ \phi^{4}(q) = (2K/\pi)^{2}$ could be expressed in the form of a generalized hypergeometric series.

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Modular Equations and Approximations to π(PI): Part 1

In this post we will discuss Ramanujan's classic paper "Modular Equations and Approximations to $ \pi$" where Ramanujan offered many amazing formulas and approximations for $ \pi$ and showed us the way to create new theories of elliptic and theta functions. However the paper as written in his classic style is devoid of proofs of the most important results. The post would try to elaborate on some of the results mentioned therein.

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Ramanujan's Class Invariants

After a heavy discussion on the modular equations found by Ramanujan, we will now focus on another significant discovery of his namely "Class Invariants".

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Elementary Approach to Modular Equations: Ramanujan's Theory 7

Continuing from previous post we proceed to derive further modular equations of degree $5$ in this post. Clearly in order to establish such equation we need to establish further theta function identities. This time we establish an identity concerning Ramanujan's $\psi$ function.

Identity Concerning $\psi(q)$ of Degree $5$

We will establish the following identity $$\psi^{2}(q^{2}) - q^{2}\psi^{2}(q^{10}) = \frac{\phi(-q^{10})f(-q^{10})}{\chi(-q^{2})}\tag{1}$$

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Elementary Approach to Modular Equations: Ramanujan's Theory 6

The Fundamental Formulas

In this post we will continue our journey of modular equations and derive a host of these mostly by using Lambert series for various theta functions. The following formula (see equation $ (14)$ of this post) will be of great help here: $$\phi^{2}(-ab)\,\frac{f(a, b)}{f(-a, -b)} = 1 + 2\sum_{n = 1}^{\infty}\frac{a^{n} + b^{n}}{1 + a^{n}b^{n}}\tag{1}$$

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Elementary Approach to Modular Equations: Ramanujan's Theory 5

Quintuple Product Identity

We first establish an identity similar to Jacobi's Triple Product which involves five factors and is quite useful in establishing various other identities involving q-series and products. This was first introduced in the mathematical literature by G. N. Watson in order to prove some of Ramanujan's theorems. The quintuple product identity is given by \begin{align}&\prod_{n = 1}^{\infty}(1 - q^{n})(1 - q^{n}z)(1 - q^{n - 1}z^{-1})(1 - q^{2n - 1}z^{2})(1 - q^{2n - 1}z^{-2})\notag\\ &\,\,\,\,\,\,\,\,= \sum_{n = -\infty}^{\infty}q^{n(3n + 1)/2}(z^{3n} - z^{-3n - 1})\notag\end{align}

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Elementary Approach to Modular Equations: Ramanujan's Theory 4

Lambert Series

In this post we will focus our attention on series of the form: $$\sum_{n = 0}^{\infty}a_{n}\cdot\frac{q^{b_n}}{1 \pm q^{c_n}}$$ which are more popularly known as Lambert Series. We will not deal with the general theorems concerning such series but will restrict ourselves to the Lambert series for the theta functions and study some identities involving these series.

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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 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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