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.