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: 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: 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: Fourier Series

We have discussed various interesting properties of elliptic functions and related theta functions in detail in previous posts. In particular we discussed that these elliptic functions are doubly periodic. It is only natural then to seek the Fourier series corresponding to these functions. However in this case we will use only the real periods to expand these functions in a Fourier series. It turns out that the Fourier expansions provide us many important identities which can be used in surprisingly many ways to connect to number theory.

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Elliptic Functions: Theta Function Identities

In this post we will prove some theta function identities. We will try to pass from an existing identity between the elliptic functions to a corresponding identity between theta functions. Sometimes we will also establish identities which follow quite obviously from the series or product expansions of the theta functions. Most of the times we will also make use of the Liouville's theorem that any doubly periodic entire function is a constant.

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Elliptic Functions: Theta Functions Contd.

The values of theta functions at the point $ z = 0$ are themselves very important and surprisingly have deep connections with number theory. For example consider $$\theta_{3}(q) = \sum_{n = -\infty}^{\infty} q^{n^{2}}$$ Then clearly $$\theta_{3}^{2}(q) = \sum_{i, j = -\infty}^{\infty} q^{i^{2} + j^{2}} = \sum_{n = 0}^{\infty} r(n)q^{n}$$ where $ r(n) = r_{2}(n)$ represents the number of ways in which integer $ n$ can be expressed as the sum of two squares (here we count order as well as sign separately).

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Elliptic Functions: Genesis of Theta Functions

We have seen in the last post that the elliptic functions can be expressed in the form of infinite product and these products look actually like ratio of two infinite products. We wish to consider these products (which are more commonly known as theta functions) in more detail in this post.

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Elliptic Functions: Infinite Products

Preliminary Results

Let us consider the ascending Landen sequence of moduli $$\cdots < k_{-n} < k_{-(n - 1)} < \cdots < k_{-2} < k_{-1} < k_{0} = k < k_{1} < k_{2} < \cdots < k_{n} < \cdots$$ where $$k_{n + 1} = \frac{2\sqrt{k_{n}}}{1 + k_{n}},\,\, k_{n} = \frac{1 - k_{n + 1}'}{1 + k_{n + 1}'}$$ Then it can be checked easily that the sequence of complementary moduli in reverse order $$\cdots < k_{n}' < k_{n - 1}' < \cdots < k_{2}' < k_{1}' < k_{0}' = k' < k_{-1}' < k_{-2}' < \cdots < k_{-n}' < \cdots$$ also forms an ascending Landen sequence.

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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: Double Periodicity Contd.

In the previous post, we had some general discussion on the periodicity and came to the conclusion that an analytic function can have at most two independent periods and in that case the ratio of periods can not be real. It was also established that if $ \omega_{1}$ and $ \omega_{2}$ are two independent periods then any period $ \omega$ can be expressed as $ \omega = m\omega_{1} + n\omega_{2}$ where $ m, n$ are integers.

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Elliptic Functions: Double Periodicity

We have seen in the previous post that the elliptic functions have at least two distinct periods. In the current post we shall discuss in details the ramifications of the double periodicity and will get a flavor of some of the methods of analytic function theory. An outline of the topics we would specifically discuss is provided below:

• Periodicity in general - a function can have at most two most periods and in case it has two periods the ratio between periods can not be real.
• Lattices and period parallelogram - position of zeroes and poles of elliptic functions.
• Liouville's theorem on elliptic 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: Addition Formulas

After having dealt with the basic properties of elliptic functions in the previous post we shall now focus on the addition formulas for them. These are used to express the functions of sum of two arguments in terms of functions of each argument separately. The additions formulas are algebraic in nature and in fact, in general any function with an algebraic addition formula is necessarily an elliptic function or a limiting case of it. We will not prove this general result here as it requires the use of theory of functions, but we shall be content to derive the formulas for the specific elliptic function which we are considering here.

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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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