elliptic integral

An elliptic integral is an integral of the form ∫R(t, √P(t)) dt, where P(t) is a polynomial of degree three or four and R is a rational function. These integrals cannot, in general, be evaluated in terms of elementary functions — they define genuinely new transcendental quantities.

The complete elliptic integral of the first kind, K(k), appears whenever you write down the exact period of a pendulum swinging through a finite angle. The standard result is T = 4√(l/g) · K(sin(θ₀/2)), where θ₀ is the maximum angle. For small angles K reduces to π/2, recovering the familiar T = 2π√(l/g). For large angles K grows without bound as θ₀ approaches π, meaning the period stretches toward infinity as the pendulum nears the top.

K(k) can be computed efficiently using the arithmetic-geometric mean (AGM): iterate aₙ₊₁ = (aₙ + bₙ)/2 and bₙ₊₁ = √(aₙbₙ) starting from a₀ = 1, b₀ = √(1 − k²). The sequences converge quadratically to a common limit M, and K(k) = π/(2M). This makes numerical evaluation fast and precise.