Jacobi elliptic functions
/ yuh-KOH-bee /
Sine and cosine are the inverses of the simplest arc-length integral, the one with a square root of (1 minus x^2). If you start instead from the elliptic arc-length integral, its inverse is a new pair of periodic functions. The Jacobi elliptic functions sn, cn and dn are exactly that generalisation of sine and cosine — periodic, but with a richer double-periodic structure controlled by a modulus.
They are defined by inverting the incomplete elliptic integral of the first kind: if u = F(phi, k), then sn(u, k) = sin phi, cn(u, k) = cos phi, and dn(u, k) = square root of (1 minus k^2 sin^2 phi). They satisfy circular-trig-like identities — sn^2 plus cn^2 = 1 and k^2 sn^2 plus dn^2 = 1 — and clean derivatives, d(sn)/du = cn times dn. When the modulus k = 0 they collapse to sn = sin, cn = cos, dn = 1; when k = 1 they degenerate into hyperbolic functions, sn = tanh, cn = dn = sech. For intermediate k they have two periods (one real, one imaginary), making them doubly periodic functions of a complex variable.
Jacobi functions give the exact solution of the nonlinear pendulum, the motion of a spinning rigid body (Euler's equations), the cnoidal and solitary waves of shallow-water and KdV theory, and oscillations in nonlinear circuits. Wherever a small-angle linearisation is too crude and the true nonlinear oscillation is needed, sn, cn and dn provide the closed-form answer that ordinary sine and cosine cannot.
The exact swing of a pendulum of amplitude theta_0 is theta(t) = 2 arcsin( k sn(omega t, k) ) with k = sin(theta_0/2); as theta_0 tends to 0, k tends to 0, sn tends to sin, and this collapses to the familiar small-angle cosine motion.
sn smoothly interpolates between ordinary sine (k = 0) and the tanh-shaped solitary-wave limit (k = 1).
There are twelve Jacobi functions in all (sn, cn, dn and their reciprocals and ratios), and the modulus convention (k versus m = k^2) again varies between references — match your software's convention.