elliptic integral,
n. a definite integral not usually evaluable in closed form by antidifferentiation. More precisely, an elliptic integral is an integral of the form
,
where R is a rational function of x and y, where y2 is a quartic polynomial in x, and where c is a fixed constant. The integral is called a complete elliptic integral if the range of integration is maximal; otherwise, it is an incomplete elliptic integral. The two most fundamental elliptic integrals are
(the complete elliptic integral of the first kind), and
(the complete elliptic integral of the second kind). For 0 < u < π/2
is an incomplete elliptic integral of the first kind. When u = π/2, this coincides with K(k). Elliptic functions can be used to evaluate the period of a pendulum with amplitude α and length L as
,
which yields the simple harmonic approximation 
for small amplitudes. Similarly, the length of the circumference of an ellipse in standard form is 4aE(e), where a is the length of the major axis, and e is the eccentricity. See arithmetic-geometric mean iteration.
,
(the complete elliptic integral of the first kind), and
(the complete elliptic integral of the second kind). For 0 < u < π/2
is an incomplete elliptic integral of the first kind. When u = π/2, this coincides with K(k). Elliptic functions can be used to evaluate the period of a pendulum with amplitude α and length L as
, for small amplitudes. Similarly, the length of the circumference of an ellipse in standard form is 4aE(e), where a is the length of the major axis, and e is the eccentricity. See arithmetic-geometric mean iteration.