Definition:Elliptic Integral of the First Kind

Complete Elliptical Integral of the First Kind
The integral:
 * $$\int \limits_0^{\pi / 2} {\frac {\mathrm{d} \phi} {\sqrt{1 - k^2 \sin^2 \phi}}}$$

is known as the Complete Elliptical Integral of the First Kind, and is a function of $$k$$, defined on the interval $$0 < k < 1$$.

It is denoted $$K \left({k}\right)$$.

Incomplete Elliptical Integral of the First Kind
The integral:


 * $$\int_0^{\phi} {\frac {\mathrm{d} \phi} {\sqrt{1 - k^2 \sin^2 \phi}}}$$

is known as the Incomplete Elliptic Integral of the First Kind, and is a function of the variables:
 * $$k$$, defined on the interval $$0 < k < 1$$;
 * $$\phi$$, defined on the interval $$0 \le \phi \le \pi / 2$$.

It is denoted $$F \left({k, \phi}\right)$$.

Note that $$F \left({k, \frac \pi 2}\right) = K \left({k}\right)$$.