Elliptic integrals were at the center of analysis at the end of 19th century. The complete elliptic integral of the first kind defined by
(1.1.1) K(k) :=
dx√ (1− x2)(1− k2x2)
is a function of the so-called modulus k2. The corresponding complete elliptic integral of the second kind is defined by
(1.1.2) E(k) :=
√ 1− k2x2 1− x2 dx.