complete elliptic integral
/ ee-LIP-tik /
Two ancient questions resist elementary calculus: how long is one full swing of a pendulum at large amplitude, and what is the exact perimeter of an ellipse? Both lead to integrals of a square root of a quartic that cannot be done with elementary functions. The complete elliptic integrals are the two named answers, evaluated over a full quarter period, that finally close these classic problems.
The complete elliptic integral of the first kind is K(k) = integral from 0 to pi/2 of d theta over square root of (1 minus k^2 sin^2 theta), and of the second kind is E(k) = integral from 0 to pi/2 of square root of (1 minus k^2 sin^2 theta) d theta. The parameter k, between 0 and 1, is called the modulus. When k = 0 both reduce to elementary values, K(0) = E(0) = pi/2; as k tends to 1, K(k) diverges logarithmically while E(1) = 1. The word complete signals that the upper limit is fixed at pi/2 — a full quarter turn — distinguishing them from the incomplete versions where the upper limit is a free variable.
K and E are the exact tools for the large-amplitude pendulum (its period is 4 over the natural frequency times K(sin of half the amplitude)), the circumference of an ellipse (involving E), the magnetic field of a current loop, the buckling of beams, and the arc length of many curves. They are computed extremely fast by the arithmetic-geometric mean, so although elliptic integrals are non-elementary they are no harder to obtain numerically than a sine or a logarithm.
A pendulum released from 90 degrees has half-amplitude 45 degrees, so k = sin 45 degrees is approximately 0.707 and K(k) is approximately 1.854. Its period is therefore about 1.18 times the small-angle period 2 pi over the natural frequency.
The small-angle formula underestimates a real pendulum's period; K(k) supplies the exact large-amplitude correction.
Beware notation: some books use the modulus k, others the parameter m = k^2, so K(0.5) can mean two different numbers — always check whether the argument is k or m.