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.

A brief outline of the approach is as follows:

• Elliptic functions as inverses of the elliptic integrals
• Fundamental equations satisfied by elliptic functions
• Addition Formulas
• Extension to complex variables
• Double Periodicity Properties
• Landen's Transformation
• Expansion into Infinite products
• Genesis of theta functions

We will make minimal use of the methods of "theory of functions of complex variables" because these methods are very indirect and way too powerful in nature. Wherever possible we will prefer to use direct methods to establish any results.

Elliptic Functions as Inverses of Elliptic Integrals

Let us start with the elliptic integral of the first kind defined by: $$u = F(\phi, k) = \int_{0}^{\phi} \frac{d\theta}{\sqrt{1 - k^{2}\sin^{2}\theta}}$$ Here we keep $ 0 \leq k \leq 1$ and we can clearly see that when $ k = 0$ then $ u = \phi$ and when $ k = 1$ then $$ u = \log(\sec\phi + \tan\phi)$$ so that these functions are elementary in nature. We therefore normally assume $ 0 < k < 1$. Then we see that $ u$ is a strictly increasing function of $ \phi$ for all values of $ \phi$. (In case $ k = 1, u$ is a strictly increasing function of $ \phi$ only for $ -\pi / 2 < \phi < \pi / 2$.) Therefore we have an inverse function which relates $ \phi$ to $ u$. This is denoted by $ \text{am}\, u = \phi$ and we say that $ \phi$ is the amplitude of $ u$.  Note that this amplitude is actually also dependent on the parameter $ k$ which is normally called the modulus.

We can now define the elliptic functions $ \text{sn}, \text{cn}, \text{dn}$ as follows: \begin{align} \text{sn}(u, k) &= \sin\phi = \sin \text{am}\,u\notag\\ \text{cn}(u, k) &= \cos\phi = \cos \text{am}\,u\notag\\ \text{dn}(u, k) &= \sqrt{1 - k^{2}\sin^{2}\phi}\notag \end{align} When there is no confusion then we normally don't write the parameter $ k$ explicitly. We next need to see range of values of $ u$ for which the above definition is valid. To this end we define: $$K = K(k) = \int_{0}^{\pi / 2} \frac{d\theta}{\sqrt{1 - k^{2}\sin^{2}\theta}}$$ and note that if $ \phi = n \cdot \pi / 2$ ($ n$ being an integer) then $$\int_{0}^{\phi} \frac{d\theta}{\sqrt{1 - k^{2}\sin^{2}\theta}} = n \cdot K$$ so that the range of values of $ u$ is $ (-\infty, \infty)$.

Also we need to note that \begin{align} F(\phi + 2\pi, k) &= \int_{0}^{\phi + 2\pi} \frac{d\theta}{\sqrt{1 - k^{2}\sin^{2}\theta}}\notag\\ &= \int_{0}^{\phi}\frac{d\theta}{\sqrt{1 - k^{2}\sin^{2}\theta}} + \int_{\phi}^{\phi + 2\pi}\frac{d\theta}{\sqrt{1 - k^{2}\sin^{2}\theta}}\notag\\ &= \int_{0}^{\phi}\frac{d\theta}{\sqrt{1 - k^{2}\sin^{2}\theta}} + \int_{0}^{2\pi}\frac{d\theta}{\sqrt{1 - k^{2}\sin^{2}\theta}} = F(\phi, k) + 4K\notag\\ &= u + 4K\notag \end{align} Therefore we have $$\text{am}(u + 4K) = \phi + 2\pi = \text{am}\, u + 2\pi$$ and so we can easily see that $$\text{sn}(u + 4K) = \sin\text{am}(u + 4K) = \sin(\text{am}\,u + 2\pi) = \sin\text{am}\,u = \text{sn}\,u$$ Similarly we can see that \begin{align} \text{cn}(u + 4K) &= \text{cn}\,u\notag\\ \text{dn}(u + 2K) &= \text{dn}\,u\notag\\ \text{sn}(u + 2K) &= -\text{sn}\,u\notag\\ \text{cn}(u + 2K) &= -\text{cn}\,u\notag\\ \end{align} It is now clear that the functions $ \text{sn}\,u, \text{cn}\,u, \text{dn}\,u$ are periodic with periods $ 4K, 4K, 2K$ respectively.

We note that the common values of the elliptic functions: \begin{align} \text{sn}(0, k) &= 0, \text{cn}(0, k) = 1, \text{dn}(0, k) = 1\notag\\ \text{sn}(K, k) &= 1, \text{cn}(K, k) = 0, \text{dn}(K, k) = \sqrt{1 - k^{2}} = k'\notag \end{align} where $ k'$ is the complementary modulus.

From the definitions we get the fundamental properties: \begin{align} \text{cn}^{2}\,u + \text{sn}^{2}\,u &= 1\notag\\ \text{dn}^{2}\,u + k^{2}\text{sn}^{2}\,u &= 1\notag \end{align} and $ 0 < \text{dn}\,u \leq 1$ for all values of $ u$.

Again we note that $ F(-\phi, k) = -F(\phi, k)$ so that $ \phi = \text{am}\,u$ is an odd function of $ u$ and therefore: \begin{align} \text{sn}(-u) &= -\text{sn}\,u\notag\\ \text{cn}(-u) &= \text{cn}\,u\notag\\ \text{dn}(-u) &= \text{dn}\,u\notag \end{align} Like the trigonometric functions we define functions which are quotients and reciprocals of the basic elliptic functions: \begin{align} \text{ns}\,u &= \frac{1}{\text{sn}\,u},\,\, \text{nc}\,u = \frac{1}{\text{cn}\,u},\,\, \text{nd}\,u = \frac{1}{\text{dn}\,u}\notag\\ \text{sc}\,u &= \frac{\text{sn}\,u}{\text{cn}\,u},\,\, \text{cd}\,u = \frac{\text{cn}\,u}{\text{dn}\,u},\,\, \text{sd}\,u = \frac{\text{sn}\,u}{\text{dn}\,u}\notag\\ \text{cs}\,u &= \frac{\text{cn}\,u}{\text{sn}\,u},\,\, \text{dc}\,u = \frac{\text{dn}\,u}{\text{cn}\,u},\,\, \text{ds}\,u = \frac{\text{dn}\,u}{\text{sn}\,u}\notag \end{align} It is easy to see now that \begin{align} \frac{d}{du}(\text{sn}\,u) &= \frac{d}{d\phi}(\sin\phi)\frac{d\phi}{du} = \frac{\cos\phi}{du / d\phi} = \cos\phi\sqrt{1 - \sin^{2}\phi} = \text{cn}\,u\,\text{dn}\,u\notag\\ \frac{d}{du}(\text{cn}\,u) &= \frac{d}{d\phi}(\cos\phi)\frac{d\phi}{du} = \frac{-\sin\phi}{du / d\phi} = -\sin\phi\sqrt{1 - \sin^{2}\phi} = -\text{sn}\,u\,\text{dn}\,u\notag\\ \frac{d}{du}(\text{dn}\,u) &= \frac{d}{d\phi}(\sqrt{1 - k^{2}\sin^{2}\phi})\frac{d\phi}{du} = \frac{-k^{2}\sin\phi\cos\phi}{(\sqrt{1 - k^{2}\sin^{2}\phi})du / d\phi}\notag\\ &= -k^{2}\sin\phi\cos\phi = -k^{2}\text{sn}\,u\,\text{cn}\,u\notag \end{align} We can now summarize the elementary properties of the elliptic functions as follows: $$\text{sn}(0, k) = 0, \text{cn}(0, k) = 1, \text{dn}(0, k) = 1\tag{1}$$ $$\text{sn}(K, k) = 1, \text{cn}(K, k) = 0, \text{dn}(K, k) = k'\tag{2}$$ $$\boxed{\begin{align} \text{sn}(-u, k) &= -\text{sn}(u, k)\notag\\ \text{cn}(-u, k) &= \text{cn}(u, k)\notag\\ \text{dn}(-u, k) &= \text{dn}(u, k)\notag \end{align}}\tag{3}$$ $$\boxed{\begin{align} \text{cn}^{2}(u, k) + \text{sn}^{2}(u, k) &= 1\notag\\ \text{dn}^{2}(u, k) + k^{2}\text{sn}^{2}(u, k) &= 1\notag \end{align}}\tag{4}$$ $$\boxed{\begin{align} -1 &\leq \text{sn}(u, k) \leq 1\notag\\ -1 &\leq \text{cn}(u, k) \leq 1\notag\\ 0 &< \text{dn}(u, k) \leq 1\notag \end{align}}\tag{5}$$ $$\boxed{\begin{align} \text{sn}(u + 2K, k) &= -\text{sn}(u, k)\notag\\ \text{cn}(u + 2K, k) &= -\text{cn}(u, k)\notag\\ \text{dn}(u + 2K, k) &= \text{dn}(u, k)\notag \end{align}}\tag{6}$$ $$\boxed{\begin{align} \text{sn}(u + 4K, k) &= \text{sn}(u, k)\notag\\ \text{cn}(u + 4K, k) &= \text{cn}(u, k)\notag\\ \text{dn}(u + 4K, k) &= \text{dn}(u, k)\notag \end{align}}\tag{7}$$ $$\boxed{\begin{align} \frac{d}{du}(\text{sn}(u, k)) &= \text{cn}(u, k)\,\text{dn}(u, k)\notag\\ \frac{d}{du}(\text{cn}(u, k)) &= -\text{sn}(u, k)\,\text{dn}(u, k)\notag\\ \frac{d}{du}(\text{dn}(u, k)) &= -k^{2}\text{sn}(u, k)\,\text{cn}(u, k)\notag \end{align}}\tag{8}$$
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