Special functions
The sine function can be defined by inverting an integral: \( z = \int_0^w\frac{dt}{\sqrt{1 - t^2}} \) gives \( z = \arcsin w \), so \( w = \sin z \). Put an extra factor under the root, \( \sqrt{(1 - t^2)(1 - k^2t^2)} \), and the same trick produces the Jacobi elliptic functions \(\operatorname{sn}\), \(\operatorname{cn}\), \(\operatorname{dn}\). They behave like sine and cosine, with addition theorems, derivative rules and periods, but with one spectacular difference: in the complex plane they have two independent periods. And they are not exotic: the exact motion of a pendulum is described by them.
The number \(k\) is the modulus. The name "elliptic" comes from the arc length of an ellipse, which leads to integrals of this kind (of the second kind, strictly speaking; see the post on arc length). Such integrals are not elementary: no combination of logarithms, trigonometric and algebraic functions has derivative \( \frac{1}{\sqrt{(1 - t^2)(1 - k^2t^2)}} \). So instead of trying to evaluate them, we accept them as new functions.
(Form (2) follows from (1) by \( dt = \cos\theta\,d\theta \) and \( \sqrt{1 - t^2} = \cos\theta \). The singularity at \(t = 1\) in (1) is integrable, like \(\frac{1}{\sqrt{1 - t}}\), which is why \(w = 1\) is allowed.)
The key idea, due to Abel and Jacobi: study the integral not as a function of its upper limit, but its inverse, the upper limit as a function of the value \(z\). This is exactly how \(\sin\) arises from \(\arcsin\), and the inverse functions turn out to be far nicer than the integrals.
Since \(\operatorname{sn}z = \sin\varphi\), we have \( \operatorname{cn}z = \cos(\operatorname{am}z) \) and \( \operatorname{dn}z = \sqrt{1 - k^2\sin^2\varphi} = \frac{d\varphi}{dz} \). The square roots are taken with \( \operatorname{cn}0 = \operatorname{dn}0 = 1 \), and then continued continuously; so \(\operatorname{cn}\) does become negative (just as \(\cos\) is \(+\sqrt{1 - \sin^2}\) only on part of its domain). In this sense, the basic identities are
\[ \operatorname{sn}^2z + \operatorname{cn}^2z = 1, \qquad k^2\operatorname{sn}^2z + \operatorname{dn}^2z = 1. \]| \(k = 0\) | \(k = 1\) | |
|---|---|---|
| integral (1) | \( \int_0^w\frac{dt}{\sqrt{1 - t^2}} = \arcsin w \) | \( \int_0^w\frac{dt}{1 - t^2} = \operatorname{artanh}w \) |
| \(\operatorname{sn}z\) | \(\sin z\) | \(\tanh z\) |
| \(\operatorname{cn}z\) | \(\cos z\) | \( \frac{1}{\cosh z} \) |
| \(\operatorname{dn}z\) | \(1\) | \( \frac{1}{\cosh z} \) |
| \(K\) | \(\frac\pi2\) | \(\infty\) |
So the elliptic functions interpolate between the trigonometric functions (\(k = 0\)) and the hyperbolic functions (\(k = 1\)). As \(k\) grows from \(0\) to \(1\), the period \(4K\) grows from \(2\pi\) to infinity, and the graph of \(\operatorname{sn}\) gets flatter at the top, spending longer near \(\pm1\).
The real period. \(K\) is the value of the integral up to \(w = 1\): the "quarter period", the analogue of \(\frac\pi2\). As \(z\) goes from \(0\) to \(K\), \(\operatorname{sn}\) rises from \(0\) to \(1\); then, as for the sine, it comes back down, goes to \(-1\) at \(3K\), and returns to \(0\) at \(4K\). In terms of the amplitude: \( \operatorname{am}(z + 2K) = \operatorname{am}z + \pi \), which flips the sign of \(\sin\) and \(\cos\) (so they need \(4K\)) but not of \(\sqrt{1 - k^2\sin^2}\) (so \(\operatorname{dn}\) needs only \(2K\)).
The imaginary period is the new feature. In the complex plane, \(\operatorname{sn}\) also repeats when \(z\) moves by \(2iK'\). The values of \(\operatorname{sn}\) on the whole plane are thus determined by its values on one period parallelogram (here a rectangle with sides \(4K\) and \(2K'\)), and the plane is tiled by copies of it. A function with two independent periods, analytic except for poles, is called elliptic; Jacobi's functions are the classical examples. (Theorem of Jacobi: a nonconstant analytic function cannot have three independent periods, and a nonconstant elliptic function must have poles; \(\operatorname{sn}\) has simple poles at \(iK'\) and \(2K + iK'\) in each parallelogram.)
Set \(k = 0\): \(\operatorname{dn} = 1\), \(\Delta = 1\), and these become \( \sin(z_1 + z_2) = \sin z_1\cos z_2 + \cos z_1\sin z_2 \) and the cosine formula. The elliptic versions have the extra \(\operatorname{dn}\) factors and the denominator. (Check at \(k = 0.6\), \(z_1 = 0.4\), \(z_2 = 0.9\): both sides of the \(\operatorname{sn}\) formula equal \(0.93416\).)
Why the derivatives look like this. By (1), \( \frac{dz}{dw} = \frac{1}{\sqrt{(1 - w^2)(1 - k^2w^2)}} \), so
\[ \frac{d}{dz}\operatorname{sn}z = \sqrt{(1 - \operatorname{sn}^2z)(1 - k^2\operatorname{sn}^2z)} = \operatorname{cn}z\operatorname{dn}z. \]The other two follow by differentiating \( \operatorname{cn}^2 = 1 - \operatorname{sn}^2 \) and \( \operatorname{dn}^2 = 1 - k^2\operatorname{sn}^2 \). Note how they mirror \( (\sin)' = \cos \), \( (\cos)' = -\sin \), with the third function \(\operatorname{dn}\) as an extra factor. In particular \( \operatorname{am}'z = \operatorname{dn}z \).
For small swings, a pendulum of length \(L\) is a harmonic oscillator with period \( T_0 = 2\pi\sqrt{L/g} \). For large swings the equation \( \ddot\theta + \frac gL\sin\theta = 0 \) is nonlinear, and its exact solution uses elliptic functions. With amplitude \(\theta_0\) and modulus \( k = \sin\frac{\theta_0}{2} \):
\[ \sin\frac{\theta(t)}{2} = k\,\operatorname{sn}\Big(\sqrt{\tfrac gL}\,t + K,\ k\Big), \qquad T = 4K(k)\sqrt{\frac Lg}. \](Energy conservation gives \( \dot\theta^2 = \frac{2g}{L}(\cos\theta - \cos\theta_0) \); the substitution \( \sin\frac\theta2 = k\sin\phi \) turns the time integral into exactly form (2).)
| Concept | Formula |
|---|---|
| Elliptic integral (1st kind) | \( z = \int_0^w\frac{dt}{\sqrt{(1 - t^2)(1 - k^2t^2)}} = \int_0^\varphi\frac{d\theta}{\sqrt{1 - k^2\sin^2\theta}} \) |
| Inverse functions | \( \varphi = \operatorname{am}z \), \( \operatorname{sn}z = \sin\operatorname{am}z = w \), \( \operatorname{cn}z = \cos\operatorname{am}z \), \( \operatorname{dn}z = \sqrt{1 - k^2\operatorname{sn}^2z} \) |
| Identities | \( \operatorname{sn}^2 + \operatorname{cn}^2 = 1 \), \( k^2\operatorname{sn}^2 + \operatorname{dn}^2 = 1 \) |
| Quarter periods | \( K = K(k) \), \( K' = K(k') \), \( k' = \sqrt{1 - k^2} \) |
| Periods | sn: \(4K, 2iK'\); cn: \(4K, 2K + 2iK'\); dn: \(2K, 4iK'\) |
| Derivatives | \( \operatorname{sn}' = \operatorname{cn}\operatorname{dn} \), \( \operatorname{cn}' = -\operatorname{sn}\operatorname{dn} \), \( \operatorname{dn}' = -k^2\operatorname{sn}\operatorname{cn} \) |
| Limits | \(k = 0\): \(\sin, \cos, 1\); \(k = 1\): \(\tanh, \operatorname{sech}, \operatorname{sech}\) |
Try each one before opening the solution. More solved problems from this topic are in Complex Numbers.
\(z = 0\): \(0, 1, 1\). \(z = 2K\) (amplitude \(\pi\)): \( \operatorname{sn} = 0 \), \( \operatorname{cn} = -1 \), \( \operatorname{dn} = 1 \).
The integrand in (1) is even in \(t\), so the integral is odd in \(w\); its inverse \(\operatorname{sn}\) is odd. Then \(\operatorname{cn}\) and \(\operatorname{dn}\), which depend on \(\operatorname{sn}^2\) and equal \(1\) at \(0\), are even.
The first is \( (\operatorname{cn}\operatorname{dn})^2 \) rewritten. Differentiating it: \( 2y'y'' = y'\big(-2(1 + k^2)y + 4k^2y^3\big) \); divide by \(2y'\).
With \( \operatorname{sn}K = 1 \), \( \operatorname{cn}K = 0 \), \( \operatorname{dn}K = k' \): numerator \( \operatorname{cn}z\operatorname{dn}z \), denominator \( 1 - k^2\operatorname{sn}^2z = \operatorname{dn}^2z \). So \( \operatorname{sn}(z + K) = \frac{\operatorname{cn}z}{\operatorname{dn}z} \), the analogue of \( \sin(z + \frac\pi2) = \cos z \).
\( K = \int_0^{\pi/2}(1 - k^2\sin^2\theta)^{-1/2}d\theta \approx \int_0^{\pi/2}\big(1 + \tfrac12k^2\sin^2\theta\big)d\theta = \frac\pi2\big(1 + \frac{k^2}{4}\big) \).
\( k = \sin15^\circ \approx 0.259 \). By Problem 5, \( \frac{T}{T_0} = \frac{2K}{\pi} \approx 1 + \frac{k^2}{4} \approx 1.017 \): about 1.7%.
Jacobi's elliptic functions come from one idea: invert the integral. The elliptic integral of the first kind cannot be expressed in elementary terms, but its inverse \(\operatorname{sn}\) behaves like a sine, with companions \(\operatorname{cn}\) and \(\operatorname{dn}\), addition theorems and derivative rules that reduce to the trigonometric ones at \(k = 0\). What is genuinely new appears in the complex plane: two independent periods, \(4K\) and \(2iK'\), which make these functions doubly periodic. And they describe real physics, such as the pendulum at large amplitude.
Related posts: arc length (the ellipse and elliptic integrals), the Gamma and Beta functions, and the hypergeometric function (\(K = \frac\pi2 F(\frac12, \frac12, 1, k^2)\)).