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

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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.

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

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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.

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

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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.

Ramanujan's Class Invariants

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

Elementary Approach to Modular Equations: Ramanujan's Theory 7

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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 $$\psi^{2}(q^{2}) - q^{2}\psi^{2}(q^{10}) = \frac{\phi(-q^{10})f(-q^{10})}{\chi(-q^{2})}\tag{1}$$