Comparing trigonometric and Jacobi elliptic functions

My previous post looked at Jacobi functions from a reference perspective: given a Jacobi function defined one way, how do I relate that to the same function defined another way, and how would you compute it?

This post explores the analogy between trigonometric functions and Jacobi elliptic functions.

Related basic Jacobi functions to trig functions

In the previous post we mentioned a connection between the argument u of a Jacobi function and the amplitude φ:

$u = \int_0^{\varphi} \frac{d\theta}{\sqrt{1-m\sin^2\theta}}$

We can use this to define the functions sn and cn. Leaving the dependence on m implicit, we have

$\begin{align*} \mathrm{sn}(u) &= \sin(\varphi) \\ \mathrm{cn}(u) &= \cos(\varphi) \end{align*}$

If m = 0, then u = φ and so sn and cn are exactly sine and cosine.

There’s a third Jacobi function we didn’t discuss much last time, and that’s dn. It would be understandable to expect dn might be analogous to tangent, but it’s not. The function dn is the derivative of φ with respect to u, or equivalently

$\mathrm{dn}(u) = \sqrt{1 - m \sin^2\varphi}$

The rest of the Jacobi functions

Just as there are more trig functions than just sine and cosine, there are more Jacobi functions than sn, cn, and dn. There are two ways to define the rest of the Jacobi functions: in terms of ratios, and in terms of zeros and poles.

Ratios

I wrote a blog post one time asking how many trig functions there are. The answer depends on your perspective, and I gave arguments for 1, 3, 6, and 12. For this post, lets say there are six: sin, cos, tan, sec, csc, and ctn.

One way to look at this is to say there are as many trig functions as there are ways to select distinct numerators and denominators from the set {sin, cos, 1}. So we have tan = sin/cos, csc = 1/sin, etc.

There are 12 Jacobi elliptic functions, one for each choice of distinct numerator and denominator from {sn, cn, dn, 1}. The name of a Jacobi function is two letters, denoting the numerator and denominator, where we have {s, c, d, n} abbreviating {sn, cn, dn, 1}.

For example, cs(u) = sn(u) / cn(u) and ns(u) = 1 / sn(u).

Note that to take the reciprocal of a Jacobi function, you just reverse the two letters in its name.

Zeros and poles

The twelve Jacobi functions can be classified [1] in terms of their zeros and poles over a rectangle whose sides have length equal to quarter periods. Let’s look at an analogous situation for trig functions before exploring Jacobi functions further.

Trig functions are periodic in one direction, while elliptic functions are periodic in two directions in the complex plane. A quarter period for the basic trig functions is π/2. The six trig functions take one value out of {0, 1, ∞} at 0 and different value at π/2. So we have one trig function for each of the six ways to chose an permutation of length 2 from a set of 3 elements.

In the previous post we defined the two quarter periods K and K‘. Imagine a rectangle whose corners are labeled

s = (0, 0)
c = (K, 0)
d = (K, K‘)
n = (0, K‘)

Each Jacobi function has a zero at one corner and a pole at another. The 12 Jacobi function correspond to the 12 ways to chose a permutation of two items from a set of four.

The name of a Jacobi function is two letters, the first letter corresponding to where the zero is, and the second letter corresponding to the pole. So, for example, sn has a zero at s and a pole at n. These give same names as the ratio convention above.

Identities

The Jacobi functions satisfy many identities analogous to trigonometric identities. For example, sn and cn satisfy a Pythagorean identity just like sine and cosine.

$\mathrm{sn}^2 u + \mathrm{cn}^2 u = 1$

Also, the Jacobi functions have addition theorems, though they’re more complicated than their trigonometric counterparts.

$\begin{align*} \mathrm{sn}(u + v) &= \frac{\mathrm{sn}\,u\, \mathrm{cn}\, v\, \mathrm{dn}\,v\, + \mathrm{sn}\,v\, \mathrm{cn}\, u\, \mathrm{dn}\,u\,}{1 - m\, \mathrm{sn}^2 u\, \, \mathrm{sn^2} v\,} \\ \\ \mathrm{\mathrm{cn}\,}(u + v) &= \frac{\mathrm{cn}\, u\, \mathrm{cn}\, v\, - \mathrm{sn}\,u\, \mathrm{dn}\,u\, \mathrm{sn}\,v\, \mathrm{dn}\,v\,}{1 - m\, \mathrm{sn}^2 u\, \, \mathrm{sn^2} v\,} \\ \\ \mathrm{dn}(u + v) &= \frac{\mathrm{dn}\,u\, \mathrm{dn}\,v\, - m\, \mathrm{sn}\,u\, \mathrm{cn}\, u\, \mathrm{sn}\,v\, \mathrm{cn}\, v\,}{1 - m\, \mathrm{sn}^2 u\, \, \mathrm{sn^2} v\,} \end{align*}$

Derivatives

The derivatives of the basic Jacobi functions are given below.

$\begin{align*} \mathrm{sn}'(u) &= \mathrm{cn}(u)\, \mathrm{dn}(u) \\ \\ \mathrm{cn}'(u) &= -\mathrm{sn}(u) \,\mathrm{dn}(u) \\ \\ \mathrm{dn}'(u) &= -m\,\mathrm{sn}(u)\, \mathrm{cn}(u) \\ \end{align*}$

Note that the implicit parameter m makes an appearance in the derivative of dn. We will also need the complementary parameter m‘ = 1 – m.

The derivatives of all Jacobi functions are summarized in the table below.

$\begin{table} \centering \begin{tabular}{l|rrrr} \multicolumn{1}{l}{} & s & n & d & c \\ \cline{2-5} s & & dn cn & nd cd & nc dc \\ n & $-$ ds cs & & $m$ sd cd & sc dc \\ d & $-$ ns cs & $-m$ sn cn & & $m'$ sc nc \\ c & $-$ ns ds & $-$ sn dn & $-m'$ sd nd & \end{tabular} \end{table}$

The derivatives of the basic Jacobi functions resemble those of trig functions. They may look more complicated at first, but they’re actually more regular. You could remember them all by observing the patterns pointed out below.

Let w, x, y, z be any permutation of {s, n, d, c}. Then the derivative of wx is proportional to yx zx. That is, the derivative of every Jacobi function f is a constant times two other Jacobi functions. The names of these two functions both end in the same letter as the name of f, and the initial letters are the two letters not in the name of f.

The proportionality constants also follow a pattern. The sign is positive if and only if the letters in the name of f appear in order in the sequence s, n, d, c. Here’s a table of just the constants.

$\begin{table} \centering \begin{tabular}{l|rrrr} \multicolumn{1}{l}{} & s & n & d & c \\ \cline{2-5} s & & 1 & 1 & 1 \\ n & $-1$ & & $m$ & 1 \\ d & $-1$ & $-m$ & & $m'$ \\ c & $-1$ & $-1$ & $-m'$ & \end{tabular} \end{table}$

Note that the table is skew symmetric, i.e. its transpose is its negative.

[1] An elliptic function is determined, up to a constant multiple, by its periods, zeros, and poles. So not only do the Jacobi functions have the pattern of zeros and poles described here, these patterns uniquely determine the Jacob functions, modulo a constant. For (singly) periodic functions, the period, zeros, and poles do not uniquely determine the function. So the discussion of zeros and poles of trig functions is included for comparison, but it does not define the trig functions.