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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