incomplete elliptic integral

The complete elliptic integrals answer 'how long is a whole quarter swing'. But often you want the partial answer — how far along the pendulum has gone after a given angle, or the arc length of an ellipse from the top down to some point. The incomplete elliptic integral keeps the upper limit free, so it tracks the accumulating quantity as a function of where you stop.

The incomplete elliptic integral of the first kind is F(phi, k) = integral from 0 to phi of d theta over square root of (1 minus k^2 sin^2 theta), and of the second kind is E(phi, k) = integral from 0 to phi of square root of (1 minus k^2 sin^2 theta) d theta. The new variable phi is the amplitude (the upper limit); the modulus k still controls the eccentricity of the square root. Setting phi = pi/2 recovers the complete integrals: F(pi/2, k) = K(k) and E(pi/2, k) = E(k). These are the canonical Legendre forms into which any elliptic integral — any integral of a rational function of a variable and the square root of a cubic or quartic — can be reduced by standard substitutions.

Incomplete elliptic integrals describe partial arc lengths, the time-to-angle relation of a finite pendulum swing, the trajectory of a charged particle, and the inverse relationship that defines the Jacobi elliptic functions: sn, cn and dn are literally the inverse of F(phi, k) with respect to its amplitude. Reducing a messy square-root integral to Legendre's F and E form is the central computational skill, after which fast standard routines finish the job.