Elementary Approach to Modular Equations: Ramanujan's Theory 3

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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 $${}_{2}F_{1}\left(\frac{1}{2}, \frac{1}{2}; 1; 1 - \left(\frac{1 - x}{1 + x}\right)^{2}\right) = (1 + x)\,{}_{2}F_{1}\left(\frac{1}{2}, \frac{1}{2}; 1; x^{2}\right)$$

Elementary Approach to Modular Equations: Ramanujan's Theory 2

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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.

Elementary Approach to Modular Equations: Ramanujan's Theory 1

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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.

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

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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}

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

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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 $$l = k^{p}\prod_{s = 1}^{(p - 1)/2}\text{sn}^{4}(K - 4s\omega)$$

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

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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).

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

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

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

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

Elementary Approach to Modular Equations: Hypergeometric Series 2

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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 $$F\left(a, b; a + b + \frac{1}{2}; 4x(1 - x)\right) = F\left(2a, 2b; a + b + \frac{1}{2}; x\right)$$

Elementary Approach to Modular Equations: Hypergeometric Series 1

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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 to expected to have some background on elliptic functions and theta functions as presented in my previous series of posts (here and here).

The Mysterious Rank (of a Matrix) Demystified

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Let us start with the following system of linear equations (written in matrix form):
$$ AX = B\text{ where } A = \begin{bmatrix} a_{11} & a_{12} & \cdots & a_{1n} \\ a_{21} & a_{22} & \cdots & a_{2n} \\ \cdots & \cdots & \cdots & \cdots \\ a_{m1} & a_{m2} & \cdots & a_{mn} \end{bmatrix},\, X = \begin{bmatrix} x_{1} \\ x_{2} \\ \cdots \\ x_{n} \end{bmatrix},\, B = \begin{bmatrix} b_{1} \\ b_{2} \\ \cdots \\ b_{m} \end{bmatrix}$$ The augmented matrix of the system is: $$\tilde{A} = \begin{bmatrix} a_{11} & a_{12} & \cdots & a_{1n} & b_{1} \\ a_{21} & a_{22} & \cdots & a_{2n} & b_{2} \\ \cdots & \cdots & \cdots & \cdots & \cdots \\ a_{m1} & a_{m2} & \cdots & a_{mn} & b_{m} \end{bmatrix}$$

The Mysterious Rank (of a Matrix): Elementary Row Operations

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In the previous post we formed notions of matrix, determinants, rank of a matrix and system of linear equations. We also mentioned the fundamental theorem on solution of system of linear equations which tells us the importance of rank in deciding the nature of solution of such systems of equations. It is now time to establish the theorem and to do that we start by defining a systematic method of solving such systems of linear equations.

The Mysterious Rank (of a Matrix)

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In this post I will be talking about the simple and beautiful concept of matrices. In particular I will discuss about the rank of a matrix and its usage in solving system of linear equations. In order to keep the post to a reasonable length I will not dwell upon the usual operations on matrices and assume that the reader is familiar with them, but a paragraph or two about these operations definitely makes sense.

Field Automorphisms: A Nice Touch on "Ambiguity"

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I am sure that the title of the post is going to confuse many readers, but for lack of a better title please be content with it. Rest assured that the post itself is quite unambiguous. We would like to throw some light on the a special kind of ambiguity which we find in mathematical systems (here mainly in algebra).

A Taste of Modern Algebra: Remainder Theorem for Polynomials

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Introduction

In high school curriculum we are taught the "Remainder Theorem" as one of the important results of Algebra. The theorem and its proof are quite simple, but the practical application and theoretical ramifications of this result are quite interesting. To recall, the theorem is stated as follows:

Remainder Theorem: If $ f(x) = a_{0}x^{n} + a_{1}x^{n - 1} + \cdots + a_{n - 1}x + a_{n}$ is a polynomial with real coefficients then the remainder obtained on dividing $ f(x)$ by $ (x - a)$ is $ f(a)$.

Continuous Functions on a Closed Interval: Uniform Continuity

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Uniform Continuity vs. Continuity

We have discussed some very useful properties of continuous functions in the last few posts. In the current post we will focus on another property called "uniform continuity". To understand what this is all about it first makes sense to reiterate the meaning of usual concept of continuity as we have seen earlier.

Continuous Functions on a Closed Interval: Intermediate Value Theorem

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After discussing the boundedness property of the property of the continuous functions, its time now to discuss another fundamental property of continuous functions called the Intermediate Value Theorem. This roughly means that if a continuous function takes two values $ A$ and $ B$ then it takes all values between $ A$ and $ B$. Thus the values of the function also maintain a continuity starting from one value to another. This theorem has some practical applications in solving equations for example. But more than that this is the most widely advertized property of continuous functions and is mentioned in almost every calculus book (and normally without proof). Some authors contend that this is the essence of continuity. However it is not the case as there can be discontinuous functions which possess this property.

Continuous Functions on a Closed Interval: Boundedness Property

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In the previous post we had developed the concept of continuous functions and some of their local properties. It is now time to study some of the properties which apply to functions which are continuous on an interval. It turns out that the most useful and beautiful results present themselves when we study the functions defined on a closed interval. The magic goes away when the intervals under discussion are not closed. Why this happens (i.e. no magical properties for open intervals) is a further subtle point which we will not discuss rightaway.

Continuous Functions

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Continuous Curves

In the last two posts we discussed about the system of real numbers and understood how the real numbers provide a model for the geometrical notion of a line which is continuous without any gaps. In this post and a few forthcoming ones we will try to create a model for continuous curves. Readers are already familiar with various continuous curves for example, circle, parabola and conic sections in general. The intuitive notion about the continuity of a curve is that we can draw in on a paper without lifting pen. This fact is so intuitive that many readers will feel that it is too much of an overkill to model it using some mathematical construct.

Real Numbers Demystified: Completeness

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In the last post we wanted to create a system of numbers which have the following property:
If all the numbers be divided into two sets $ L, U$ such that every member of $ L$ is less than any member of $ U$ then there must be a number $ \alpha$ such that all numbers less than $ \alpha$ belong to $ L$ and all the numbers greater than $ \alpha$ belong to $ U$, the number $ \alpha$ may itself lie in one of the sets $ L, U$.

Real Numbers Demystified

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Prologue

I remember a dialogue I had with a member of the Mathematics community on Orkut about the concept of real numbers. This was around 4-5 years ago and he was one of the really intelligent members out there in that community and had just joined IIT Bombay. To protect his identity lets call him by the name X. So we have a discussion which goes something like the following:

Me: So you have been doing great solving the math problems posed by the community.
X: Sort of. I really love solving problems.
Me: Looking at the depth of your skills, I suppose you must have some idea of real numbers.
X: Oh yes, they are just below the Complex Numbers in the number system hierarchy.
Me: Yeah that's fine, but what would you tell about real numbers to a guy who does not know about the complex numbers?
X: I would say that the real numbers are "the rational numbers and the irrational numbers" taken together.
Me: And then what would say about the irrational numbers?
X: Irrationals are just those real numbers which are not rational.
Me: If you look carefully at what you said, you will notice that there is a circularity involved and you have not defined any of the terms "irrational" and "real" in context of numbers.
X: Yes I guess that you are correct, but I don't know how we can avoid that circularity. We have never been told otherwise about real numbers.

On Mathematics Education: Algebra vs. Calculus

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This time I am going to take a break from heavy use of $ \mathrm\LaTeX$ like I used to do in my earlier posts. I will focus on the education of mathematics as is being provided to students in India. My sources of information in this regard are:
  • memories from my school/college days (which are still relevant in current time)
  • interaction with current school/college students
  • mathematics books (both Indian and foreign, old and new) available on the market
The subtopic of the post should not be taken too literally, rather it reflects two approaches in the teaching of mathematics: 1) algebraic approach which I don't like and find unsuitable for any serious teaching of mathematics, and 2) the calculus approach which is the way mathematics teaching should be, but is currently not being practiced anywhere as far as the books show.

Irrationality of π(PI): Lambert’s Proof Contd.

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Irrationality of Continued Fractions

We have the following results about the irrationality of some continued fractions:
If $ a_{n}, b_{n}$ are positive integers then:
1) The infinite continued fraction $$\frac{b_{2}}{a_{2} +}\,\frac{b_{3}}{a_{3} +}\,\frac{b_{4}}{a_{4} +}\,\frac{b_{5}}{a_{5} +}\,\cdots $$ converges to an irrational value, provided that $ a_{n} \geq b_{n}$ for all values of $ n$ starting from a certain value $ n = n_{0}$.

2) The infinite continued fraction $$\frac{b_{2}}{a_{2} -}\,\frac{b_{3}}{a_{3} -}\,\frac{b_{4}}{a_{4} -}\,\frac{b_{5}}{a_{5} -}\,\cdots $$ converges to an irrational value, provided that $ a_{n} \geq b_{n} + 1$ for all values of $ n$ starting from a certain value $ n = n_{0}$ and the condition $ a_{n} > b_{n} + 1$ must hold for an infinite number of values of $ n$.

Irrationality of π(PI): Lambert's Proof

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Introduction

After mentioning about the Lambert's famous proof of irrationality of $ \pi$ in an earlier post, it is now time to give it to the readers in its entirety. I need to reiterate the fact that being a far more direct proof than the modern proofs of Ivan Niven, it is still highly neglected by modern authors and educators. The idea of the proof is really elementary but based on the concept of continued fractions which are now deleted from the high school mathematical syllabus. Why this topic is now left out is still unclear to me. One reason which I can guess of is that the manipulations of continued fractions are not so simple (compared to those of an infinite product or a series). The visible form of the continued fraction does not give any idea about its value unless we do the calculations.

Continued fraction expansion of tan(x)

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Introduction

One of the most intriguing formulas which I encountered during my 12th grade was the continued fraction expansion of $ \tan x$. It was given in a book on "Numerical Analysis" and was offered as an example of a formula which converges very fast (like something comparable to the Maclaurin series for $ \sin x$ and $ \cos x$). However like the usual practice followed in mathematics textbooks it was offered without any proof or any context. I checked the formula using calculator and was amazed with its speed of convergence. Just 10 terms were enough to give the result with an accuracy up to 10 decimals.

Elliptic Functions: Fourier Series

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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.

Elliptic Functions: Theta Function Identities

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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.

Elliptic Functions: Theta Functions Contd.

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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).

Elliptic Functions: Genesis of Theta Functions

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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.

Elliptic Functions: Infinite Products

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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.

Elliptic Functions: Landen's Transformation

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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.

Elliptic Functions: Double Periodicity Contd.

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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.

Elliptic Functions: Double Periodicity

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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

Elliptic Functions: Complex Variables

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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.

Elliptic Functions: Addition Formulas

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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.

Elliptic Functions: Introduction

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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.